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A V C T O 



Ac a plurimis praeccclcntium edifionum 
mendis expurgata. 



V E N E T I I S, 


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r^IO^Z/M PZllMII.yi/, ac P R IV 1LE G 1 0, 










IN 1763 













HE text presented in this volume is that of the Venetian edition of 1763. 
This edition was chosen in preference to the first edition of 1758, published 
at Vienna, because, as stated on the title-page, it was the first edition (revised 
and enlarged) issued under the personal superintendence of the author. 

In the English translation, an endeavour has been made to adhere as 
closely as possible to a literal rendering of the Latin ; except that the some- 
what lengthy and complicated sentences have been broken up. This has 
made necessary slight changes of meaning in several of the connecting words. This will be 
noted especially with regard to the word " adeoque ", which Boscovich uses with a variety 
of shades of meaning, from " indeed ", " also " or " further ", through " thus ", to a decided 
" therefore ", which would have been more correctly rendered by " ideoque ". There is 
only one phrase in English that can also take these various shades of meaning, viz., " and so " ; 
and this phrase, for the use of which there is some justification in the word " adeo " itself, 
has been usually employed. 

The punctuation of the Latin is that of the author. It is often misleading to a modern 
reader and even irrational ; but to have recast it would have been an onerous task and 
something characteristic of the author and his century would have been lost. 

My translation has had the advantage of a revision by Mr. A. O. Prickard, M.A., Fellow 
of New College, Oxford, whose task has been very onerous, for he has had to watch not 
only for flaws in the translation, but also for misprints in the Latin. These were necessarily 
many ; in the first place, there was only one original copy available, kindly loaned to me by 
the authorities of the Cambridge University Library ; and, as this copy could not leave 
my charge, a type-script had to be prepared from which the compositor worked, thus doub- 
ling the chance of error. Secondly, there were a large number of misprints, and even 
omissions of important words, in the original itself ; for this no discredit can be assigned to 
Boscovich ; for, in the printer's preface, we read that four presses were working at the 
same time in order to take advantage of the author's temporary presence in Venice. Further, 
owing to almost insurmountable difficulties, there have been many delays in the production 
of the present edition, causing breaks of continuity in the work of the translator and reviser ; 
which have not conduced to success. We trust, however, that no really serious faults remain. 
The short life of Boscovich, which follows next after this preface, has been written by 
Dr. Branislav Petronievic, Professor of Philosophy at the University of Belgrade. It is to 
be regretted that, owing to want of space requiring the omission of several addenda to the 
text of the Theoria itself, a large amount of interesting material collected by Professor 
Petronievic has had to be left out. 

The financial support necessary for the production of such a costly edition as the present 
has been met mainly by the Government of the Kingdom of Serbs, Croats and Slovenes ; 
and the subsidiary expenses by some Jugo-Slavs interested in the publication. 

After the " Life," there follows an " Introduction," in which I have discussed the ideas 
of Boscovich, as far as they may be gathered from the text of the Tbeoria alone ; this 
also has been cut down, those parts which are clearly presented to the reader in Boscovich's 
own Synopsis having been omitted. It is a matter of profound regret to everyone that this 
discussion comes from my pen instead of, as was originally arranged, from that of the late 
Philip E. P. Jourdain, the well-known mathematical logician ; whose untimely death threw 
into my far less capable hands the responsible duties of editorship. 

I desire to thank the authorities of the Cambridge University Library, who time after 
time over a period of five years have forwarded to me the original text of this work of 
Boscovich. Great credit is also due to the staff of Messrs. Butler & Tanner, Frome, 
for the care and skill with which they have carried out their share of the work ; and 
my special thanks for the unfailing painstaking courtesy accorded to my demands, which were 
frequently not in agreement with trade custom. 


December, 1921. 



]HE Slav world, being still in its infancy, has, despite a considerable number 
of scientific men, been unable to contribute as largely to general science 
as the other great European nations. It has, nevertheless, demonstrated 
its capacity of producing scientific works of the highest value. Above 
all, as I have elsewhere indicated," it possesses Copernicus, Lobachevski, 
Mendeljev, and Boscovich. 

In the following article, I propose to describe briefly the life of the 
Jugo-Slav, Boscovich, whose principal work is here published for the sixth time ; the first 
edition having appeared in 1758, and others in 1759, 1763, 1764, and 1765. The present 
text is from the edition of 1763, the first Venetian edition, revised and enlarged. 

On his father's side, the family of Boscovich is of purely Serbian origin, his grandfather, 
Bosko, having been an orthodox Serbian peasant of the village of Orakova in Herzegovina. 
His father, Nikola, was first a merchant in Novi Pazar (Old Serbia), but later settled in 
Dubrovnik (Ragusa, the famous republic in Southern Dalmatia), whither his father, Bosko, 
soon followed him, and where Nikola became a Roman Catholic. Pavica, Boscovich's 
mother, belonged to the Italian family of Betere, which for a century had been established 
in Dubrovnik and had become Slavonicized Bara Betere, Pavica's father, having been a 
poet of some reputation in Ragusa. 

Roger Joseph Boscovich (Rudjer Josif Boskovic', in Serbo-Croatian) was born at Ragusa 
on September i8th, 1711, and was one of the younger members of a large family. He 
received his primary and secondary education at the Jesuit College of his native town ; 
in 1725 he became a member of the Jesuit order and was sent to Rome, where from 1728 
to 1733 he studied philosophy, physics and mathematics in the Collegium Romanum. 
From 1733 to 1738 he taught rhetoric and grammar in various Jesuit schools ; he became 
Professor of mathematics in the Collegium Romanum, continuing at the same time his 
studies in theology, until in 1744 he became a priest and a member of his order. 

In 1736, Boscovich began his literary activity with the first fragment, " De Maculis 
Solaribus," of a scientific poem, " De Solis ac Lunse Defectibus " ; and almost every 
succeeding year he published at least one treatise upon some scientific or philosophic problem. 
His reputation as a mathematician was already established when he was commissioned by 
Pope Benedict XIV to examine with two other mathematicians the causes of the weakness 
in the cupola of St. Peter's at Rome. Shortly after, the same Pope commissioned him to 
consider various other problems, such as the drainage of the Pontine marshes, the regulariza- 
tion of the Tiber, and so on. In 1756, he was sent by the republic of Lucca to Vienna 
as arbiter in a dispute between Lucca and Tuscany. During this stay in Vienna, Boscovich 
was commanded by the Empress Maria Theresa to examine the building of the Imperial 
Library at Vienna and the cupola of the cathedral at Milan. But this stay in Vienna, 
which lasted until 1758, had still more important consequences ; for Boscovich found 
time there to finish his principal work, Theoria Philosophies Naturalis ; the publication 
was entrusted to a Jesuit, Father Scherffer, Boscovich having to leave Vienna, and the 
first edition appeared in 1758, followed by a second edition in the following year. With 
both of these editions, Boscovich was to some extent dissatisfied (see the remarks made 
by the printer who carried out the third edition at Venice, given in this volume on page 3) ; 
so a third edition was issued at Venice, revised, enlarged and rearranged under the author's 
personal superintendence in 1763. The revision was so extensive that as the printer 
remarks, " it ought to be considered in some measure as a first and original edition " ; 
and as such it has been taken as the basis of the translation now published. The fourth 
and fifth editions followed in 1764 and 1765. 

One of the most important tasks which Boscovich was commissioned to undertake 
was that of measuring an arc of the meridian in the Papal States. Boscovich had designed 
to take part in a Portuguese expedition to Brazil on a similar errand ; but he was per- 

" Slav Achievements in Advanced Science, London, 1917. 



suaded by Pope Benedict XIV, in 1750, to conduct, in collaboration with an English Jesuit, 
Christopher Maire, the measurements in Italy. The results of their work were published, 
in 1755, by Boscovich, in a treatise, De Litter aria Expedition^ -per Pontificiam, &c. ; this 
was translated into French under the title of Voyage astronomique et geograpbique dans 
VEtat de VEglise, in 1770. 

By the numerous scientific treatises and dissertations which he had published up to 
1759, and by his principal work, Boscovich had acquired so high a reputation in Italy, nay 
in Europe at large, that the membership of numerous academies and learned societies had 
already been conferred upon him. In 1760, Boscovich, who hitherto had been bound to 
Italy by his professorship at Rome, decided to leave that country. In this year we find 
him at Paris, where he had gone as the travelling companion of the Marquis Romagnosi. 
Although in the previous year the Jesuit order had been expelled from France, Boscovich 
had been received on the strength of his great scientific reputation. Despite this, he did not 
feel easy in Paris ; and the same year we find him in London, on a mission to vindicate 
the character of his native place, the suspicions of the British Government, that Ragusa was 
being used by France to fit out ships of war, having been aroused ; this mission he carried 
out successfully. In London he was warmly welcomed, and was made a member of the 
Royal Society. Here he published his work, De Solis ac Lunce defectibus, dedicating it to 
the Royal Society. Later, he was commissioned by the Royal Society to proceed to Cali- 
fornia to observe the transit of Venus ; but, as he was unwilling to go, the Society sent 
him to Constantinople for the same purpose. He did not, however, arrive in time to 
make the observation ; and, when he did arrive, he fell ill and was forced to remain at 
Constantinople for seven months. He left that city in company with the English ambas- 
sador, Porter, and, after a journey through Thrace, Bulgaria, and Moldavia, he arrived 
finally at Warsaw, in Poland ; here he remained for a time as the guest of the family of 
PoniatowsM. In 1762, he returned from Warsaw to Rome by way of Silesia and Austria. 
The first part of this long journey has been described by Boscovich himself in his Giornale 
di un viaggio da Constantinopoli in Polonia the original of which was not published until 
1784, although a French translation had appeared in 1772, and a German translation 
in 1779. 

Shortly after his return to Rome, Boscovich was appointed to a chair at the University 
of Pavia ; but his stay there was not of long duration. Already, in 1764, the building 
of the observatory of Brera had been begun at Milan according to the plans of Boscovich ; 
and in 1770, Boscovich was appointed its director. Unfortunately, only two years later 
he was deprived of office by the Austrian Government which, in a controversy between 
Boscovich and another astronomer of the observatory, the Jesuit Lagrange, took the part 
of his opponent. The position of Boscovich was still further complicated by the disbanding 
of his company ; for, by the decree of Clement V, the Order of Jesus had been suppressed in 
1773. In the same year Boscovich, now free for the second time, again visited Paris, where 
he was cordially received in official circles. The French Government appointed him director 
of " Optique Marine," with an annual salary of 8,000 francs ; and Boscovich became a 
French subject. But, as an ex- Jesuit, he was not welcomed in all scientific circles. The 
celebrated d'Alembert was his declared enemy ; on the other hand, the famous astronomer, 
Lalande, was his devoted friend and admirer. Particularly, in his controversy with Rochon 
on the priority of the discovery of the micrometer, and again in the dispute with Laplace 
about priority in the invention of a method for determining the orbits of comets, did 
the enmity felt in these scientific circles show itself. In Paris, in 1779, Boscovich 
published a new edition of his poem on eclipses, translated into French and annotated, 
under the title, Les Eclipses, dedicating the edition to the King, Louis XV. 

During this second stay in Paris, Boscovich had prepared a whole series of new works, 
which he hoped would have been published at the Royal Press. But, as the American 
War of Independence was imminent, he was forced, in 1782, to take two years' leave of 
absence, and return to Italy. He went to the house of his publisher at Bassano ; and here, 
in 1 785^ were published five volumes of his optical and astronomical works, Opera pertinentia 
ad opticam et astronomiam. 

Boscovich had planned to return through Italy from Bassano to Paris ; indeed, he left 
Bassano for Venice, Rome, Florence, and came to Milan. Here he was detained by illness 
and he was obliged to ask the French Government to extend his leave, a request that was 
willingly granted. His health, however, became worse ; and to it was added a melancholia. 
He died on February I3th, 1787. 

The great loss which Science sustained by his death has been fitly commemorated in 
the eulogium by his friend Lalande in the French Academy, of which he was a member ; 
and also in that of Francesco Ricca at Milan, and so on. But it is his native town, his 
beloved Ragusa, which has most fitly celebrated the death of the greatest of her sons 


in the eulogium of the poet, Bernardo Zamagna. " This magnificent tribute from his native 
town was entirely deserved by Boscovich, both for his scientific works, and for his love and 
work for his country. 

Boscovich had left his native country when a boy, and returned to it only once after- 
wards, when, in 1747, he passed the summer there, from June 20th to October 1st ; but 
he often intended to return. In a letter, dated May 3rd, 1774, he seeks to secure a pension 
as a member of the Jesuit College of Ragusa ; he writes : " I always hope at last to find 
my true peace in my own country and, if God permit me, to pass my old age there in 

Although Boscovich has written nothing in his own language, he understood it per- 
fectly ; as is shown by the correspondence with his sister, by certain passages in his Italian 
letters, and also by his Giornale (p. 31 ; p. 59 of the French edition). In a dispute with 
d'Alembert, who had called him an Italian, he said : " we will notice here in the first place 
that our author is a Dalmatian, and from Ragusa, not Italian ; and that is the reason why 
Marucelli, in a recent work on Italian authors, has made no mention of him." * That his 
feeling of Slav nationality was strong is proved by the tributes he pays to his native town 
and native land in his dedicatory epistle to Louis XV. 

Boscovich was at once philosopher, astronomer, physicist, mathematician, historian, 
engineer, architect, and poet. In addition, he was a diplomatist and a man of the world ; 
and yet a good Catholic and a devoted member of the Jesuit order. His friend, Lalande, 
has thus sketched his appearance and his character : " Father Boscovich was of great 
stature ; he had a noble expression, and his disposition was obliging. He accommodated 
himself with ease to the foibles of the great, with whom he came into frequent contact. 
But his temper was a trifle hasty and irascible, even to his friends at least his manner 
gave that impression but this solitary defect was compensated by all those qualities which 
make up a great man. . . . He possessed so strong a constitution that it seemed likely that 
he would have lived much longer than he actually did ; but his appetite was large, and his 
belief in the strength of his constitution hindered him from paying sufficient attention 
to the danger which always results from this." From other sources we learn that Boscovich 
had only one meal daily, dejeuner. 

Of his ability as a poet, Lalande says : " He was himself a poet like his brother, who was 
also a Jesuit. . . . Boscovich wrote verse in Latin only, but he composed with extreme ease. 
He hardly ever found himself in company without dashing off some impromptu verses to 
well-known men or charming women. To the latter he paid no other attentions, for his 
austerity was always exemplary. . . . With such talents, it is not to be wondered at that 
he was everywhere appreciated and sought after. Ministers, princes and sovereigns all 
received him with the greatest distinction. M. de Lalande witnessed this in every part 
of Italy where Boscovich accompanied him in 1765." 

Boscovich was acquainted with several languages Latin, Italian, French, as well as 
his native Serbo-Croatian, which, despite his long absence from his country, he did not 
forget. Although he had studied in Italy and passed the greater part of his life there, 
he had never penetrated to the spirit of the language, as his Italian biographer, Ricca, notices. 
His command of French was even more defective ; but in spite of this fact, French men 
of science urged him to write in French. English he did not understand, as he confessed 
in a letter to Priestley ; although he had picked up some words of polite conversation 
during his stay in London. 

His correspondence was extensive. The greater part of it has been published in 
the Memoirs de VAcademie Jougo-Slave of Zagrab, 1887 to 1912. 

" Oratio in funere R. J. Boscovichii ... a Bernardo Zamagna. 

* Voyage Astronomique, p. 750 ; also on pp. 707 seq. 

Journal des Sfavans, Fevrier, 1792, pp. 113-118. 


ALTHOUGH the title to this work to a very large extent correctly describes 
the contents, yet the argument leans less towards the explanation of a 
theory than it does towards the logical exposition of the results that must 
follow from the acceptance of certain fundamental assumptions, more or 
less generally admitted by natural philosophers of the time. The most 
important of these assumptions is the doctrine of Continuity, as enunciated 
by Leibniz. This doctrine may be shortly stated in the words : " Every- 
thing takes place by degrees " ; or, in the phrase usually employed by Boscovich : " Nothing 
happens -per saltum." The second assumption is the axiom of Impenetrability ; that is to 
say, Boscovich admits as axiomatic that no two material points can occupy the same spatial, 
or local, point simultaneously. Clerk Maxwell has characterized this assumption as " an 
unwarrantable concession to the vulgar opinion." He considered that this axiom is a 
prejudice, or prejudgment, founded on experience of bodies of sensible size. This opinion 
of Maxwell cannot however be accepted without dissection into two main heads. The 
criticism of the axiom itself would appear to carry greater weight against Boscovich than 
against other philosophers ; but the assertion that it is a prejudice is hardly warranted. 
For, Boscovich, in accepting the truth of the axiom, has no experience on which to found his 
acceptance. His material points have absolutely no magnitude ; they are Euclidean points, 
" having no parts." There is, therefore, no reason for assuming, by a sort of induction (and 
Boscovich never makes an induction without expressing the reason why such induction can 
be made), that two material points cannot occupy the same local point simultaneously ; 
that is to say, there cannot have been a prejudice in favour of the acceptance of this axiom, 
derived from experience of bodies of sensible size ; for, since the material points are non- 
extended, they do not occupy space, and cannot therefore exclude another point from 
occupying the same space. Perhaps, we should say the reason is not the same as that which 
makes it impossible for bodies of sensible size. The acceptance of the axiom by Boscovich is 
purely theoretical ; in fact, it constitutes practically the whole of the theory of Boscovich. On 
the other hand, for this very reason, there are no readily apparent grounds for the acceptance 
of the axiom ; and no serious arguments can be adduced in its favour ; Boscovich 's own 
line of argument, founded on the idea that infinite improbability comes to the same thing 
as impossibility, is given in Art. 361. Later, I will suggest the probable source from which 
Boscovich derived his idea of impenetrability as applying to points of matter, as distinct 
from impenetrability for bodies of sensible size. 

Boscovich's own idea of the merit of his work seems to have been chiefly that it met the 
requirements which, in the opinion of Newton, would constitute " a mighty advance in 
philosophy." These requirements were the " derivation, from the phenomena of Nature, 
of two or three general principles ; and the explanation of the manner in which the properties 
and actions of all corporeal things follow from these principles, even if the causes of those 
principles had not at the time been discovered." Boscovich claims in his preface to the 
first edition (Vienna, 1758) that he has gone far beyond these requirements ; in that he has 
reduced all the principles of Newton to a single principle namely, that given by his Law 
of Forces. 

The occasion that led to the writing of this work was a request, made by Father Scherffer, 
who eventually took charge of the first Vienna edition during the absence of Boscovich ; he 
suggested to Boscovich the investigation of the centre of oscillation. Boscovich applied to 
this investigation the principles which, as he himself states, " he lit upon so far back as the 
year 1745." Of these principles he had already given some indication in the dissertations 
De Viribus vivis (published in 1745), De Lege Firium in Natura existentium (1755), and 
others. While engaged on the former dissertation, he investigated the production and 
destruction of velocity in the case of impulsive action, such as occurs in direct collision. 
In this, where it is to be noted that bodies of sensible size are under consideration, Boscovich 
was led to the study of the distortion and recovery of shape which occurs on impact ; he 
came to the conclusion that, owing to this distortion and recovery of shape, there was 
produced by the impact a continuous retardation of the relative velocity during the whole 
time of impact, which was finite ; in other words, the Law of Continuity, as enunciated by 



Leibniz, was observed. It would appear that at this time (1745) Boscovich was concerned 
mainly, if not solely, with the facts of the change of velocity, and not with the causes for 
this change. The title of the dissertation, De Firibus vivis, shows however that a secondary 
consideration, of almost equal importance in the development of the Theory of Boscovich, 
also held the field. The natural philosophy of Leibniz postulated monads, without parts, 
extension or figure. In these features the monads of Leibniz were similar to the material 
points of Boscovich ; but Leibniz ascribed to his monads 1 perception and appetition in 
addition to an equivalent of inertia. They are centres of force, and the force exerted is a 
vis viva. Boscovich opposes this idea of a " living," or " lively " force ; and in this first 
dissertation we may trace the first ideas of the formulation of his own material points. 
Leibniz denies action at a distance ; with Boscovich it is the fundamental characteristic of 
a material point. 

The principles developed in the work on collisions of bodies were applied to the problem 
of the centre of oscillation. During the latter investigation Boscovich was led to a theorem 
on the mutual forces between the bodies forming a system of three ; and from this theorem 
there followed the natural explanation of a whole sequence of phenomena, mostly connected 
with the idea of a statical moment ; and his initial intention was to have published a 
dissertation on this theorem and deductions from it, as a specimen of the use and advantage 
of his principles. But all this time these principles had been developing in two directions, 
mathematically and philosophically, and by this time included the fundamental notions 
of the law of forces for material points. The essay on the centre of oscillation grew in length 
as it proceeded ; until, finally, Boscovich added to it all that he had already published on 
the subject of his principles and other matters which, as he says, " obtruded themselves on 
his notice as he was writing." The whole of this material he rearranged into a more logical 
(but unfortunately for a study of development of ideas, non-chronological) order before 

As stated by Boscovich, in Art. 164, the whole of his Theory is contained in his statement 
that : " Matter is composed of perfectly indivisible, non-extended, discrete points." To this 
assertion is conjoined the axiom that no two material points can be in the same point of 
space at the same time. As stated above, in opposition to Clerk Maxwell, this is no matter 
of prejudice. Boscovich, in Art. 361, gives his own reasons for taking this axiom as part 
of his theory. He lays it down that the number of material points is finite, whereas the 
number of local points is an infinity of three dimensions ; hence it is infinitely improbable, 
i.e., impossible, that two material points, without the action of a directive mind, should 
ever encounter one another, and thus be in the same place at the same time. He even goes 
further ; he asserts elsewhere that no material point ever returns to any point of space in 
which it has ever been before, or in which any other material point has ever been. Whether 
his arguments are sound or not, the matter does not rest on a prejudgment formed from 
experience of bodies of sensible size ; Boscovich has convinced himself by such arguments 
of the truth of the principle of Impenetrability, and lays it down as axiomatic ; and upon 
this, as one of his foundations, builds his complete theory. The consequence of this axiom 
is immediately evident ; there can be no such thing as contact between any two material 
points ; two points cannot be contiguous or, as Boscovich states, no two points of matter 
can be in mathematical contact. For, since material points have no 
dimensions, if, to form an imagery of Boscovich's argument, we take 
two little squares ABDC, CDFE to represent two points in mathema- 
tical contact along the side CD, then CD must also coincide with AB, 
and EF with CD ; that is the points which we have supposed to be 
contiguous must also be coincident. This is contrary to the axiom of 
Impenetrability ; and hence material points must be separated always O U Ir 
by a finite interval, no matter how small. This finite interval however 
has no minimum ; nor has it, on the other hand, on account of the infinity of space, any 
maximum, except under certain hypothetical circumstances which may possibly exist. 
Lastly, these points of matter float, so to speak, in an absolute void. 

Every material point is exactly like every other material point ; each is postulated to 
have an inherent propensity (determinatio) to remain in a state of rest or uniform motion in 
a straight line, whichever of these is supposed to be its initial state, so long as the point is 
not subject to some external influence. Thus it is endowed with an equivalent of inertia 
as formulated by Newton ; but as we shall see, there does not enter the Newtonian idea 
of inertia as a characteristic of mass. The propensity is akin to the characteristic ascribed 
to the monad by Leibniz ; with this difference, that it is not a symptom of activity, as with 
Leibniz, but one of inactivity. 

1 See Bertrand Russell, Philosophy of Leibniz ; especially p. 91 for connection between Boscovich and Leibniz. 


Further, according to Boscovich, there is a mutual vis between every pair of points, 
the magnitude of which depends only on the distance between them. At first sight, there 
would seem to be an incongruity in this supposition ; for, since a point has no magnitude, 
it cannot have any mass, considered as " quantity of matter " ; and therefore, if the slightest 
" force " (according to the ordinary acceptation of the term) existed between two points, 
there would be an infinite acceleration or retardation of each point relative to the other. 
If, on the other hand, we consider with Clerk Maxwell that each point of matter has a 
definite small mass, this mass must be finite, no matter how small, and not infinitesimal. 
For the mass of a point is the whole mass of a body, divided by the number of points of 
matter composing that body, which are all exactly similar ; and this number Boscovich 
asserts is finite. It follows immediately that the density of a material point must be infinite, 
since the volume is an infinitesimal of the third order, if not of an infinite order, i.e., zero. 
Now, infinite density, if not to all of us, to Boscovich at least is unimaginable. Clerk 
Maxwell, in ascribing mass to a Boscovichian point of matter, seems to have been obsessed 
by a prejudice, that very prejudice which obsesses most scientists of the present day, namely, 
that there can be no force without mass. He understood that Boscovich ascribed to each 
pair of points a mutual attraction or repulsion ; and, in consequence, prejudiced by Newton's 
Laws of Motion, he ascribed mass to a material point of Boscovich. 

This apparent incongruity, however, disappears when it is remembered that the word 
vis, as used by the mathematicians of the period of Boscovich, had many different meanings ; 
or rather that its meaning was given by the descriptive adjective that was associated with it. 
Thus we have vis viva (later associated with energy), vis mortua (the antithesis of vis viva, 
as understood by Leibniz), vis acceleratrix (acceleration), vis matrix (the real equivalent 
of force, since it varied with the mass directly), vis descensiva (moment of a weight hung at 
one end of a lever), and so on. Newton even, in enunciating his law of universal gravitation, 
apparently asserted nothing more than the fact of gravitation a propensity for approach 
according to the inverse square of the distance : and Boscovich imitates him in this. The 
mutual vires, ascribed by Boscovich to his pairs of points, are really accelerations, i.e. 
tendencies for mutual approach or recession of the two points, depending on the distance 
between the points at the time under consideration. Boscovich's own words, as given in 
Art. 9, are : " Censeo igitur bina quaecunque materise puncta determinari asque in aliis 
distantiis ad mutuum accessum, in aliis ad recessum mutuum, quam ipsam determinationem 
apello vim." The cause of this determination, or propensity, for approach or recession, 
which in the case of bodies of sensible size is more correctly called " force " (vis matrix), 
Boscovich does not seek to explain ; he merely postulates the propensities. The measures 
of these propensities, i.e., the accelerations of the relative velocities, are the ordinates of 
what is usually called his curve of forces. This is corroborated by the statement of Boscovich 
that the areas under the arcs of his curve are proportional to squares of velocities ; which 
is in accordance with the formula we should now use for the area under an " acceleration- 
space " graph (Area = J f.ds = j-r-ds = I v.dv). See Note (f) to Art. 118, where it is 

evident that the word vires, translated " forces," strictly means " accelerations ; " seejalso Art.64- 
Thus it would appear that in the Theory of Boscovich we have something totally 
different from the monads of Leibniz, which are truly centres of force. Again, although 
there are some points of similarity with the ideas of Newton, more especially in the 
postulation of an acceleration of the relative velocity of every pair of points of matter due 
to and depending upon the relative distance between them, without any endeavour to 
explain this acceleration or gravitation ; yet the Theory of Boscovich differs from that of 
Newton in being purely kinematical. His material point is defined to be without parts, 
i.e., it has no volume ; as such it can have no mass, and can exert no force, as we understand 
such terms. The sole characteristic that has a finite measure is the relative acceleration 
produced by the simultaneous existence of two points of matter ; and this acceleration 
depends solely upon the distance between them. The Newtonian idea of mass is replaced 
by something totally different ; it is a mere number, without " dimension " ; the " mass " 
of a body is simply the number of points that are combined to " form " the body. 

Each of these points, if sufficiently close together, will exert on another point of matter, 
at a relatively much greater distance from every point of the body, the same acceleration 
very approximately. Hence, if we have two small bodies A and B, situated at a distance s 
from one another (the wording of this phrase postulates that the points of each body are 
very close together as compared with the distance between the bodies) : and if the number 
of points in A and B are respectively a and b, and / is the mutual acceleration between any 
pair of material points at a distance s from one another ; then, each point of A will give to 
each point of B an acceleration /. Hence, the body A will give to each point of B, and 
therefore to the whole of B, an acceleration equal to a/. Similarly the body B will give to 


the body A an acceleration equal to bf. Similarly, if we placed a third body, C, at a distance 
j from A and B, the body A would give the body C an acceleration equal to af, and the body 
B would give the body C an acceleration equal to bf. That is, the accelerations given to a 
standard body C are proportional to the " number of points " in the bodies producing 
these accelerations ; thus, numerically, the " mass " of Boscovich comes to the same thing 
as the " mass " of Newton. Further, the acceleration given by C to the bodies A and B 
is the same for either, namely, cf ; from which it follows that all bodies have their velocities 
of fall towards the earth equally accelerated, apart from the resistance of the air ; and so on. 
But the term " force," as the cause of acceleration is not applied by Boscovich to material 
points ; nor is it used in the Newtonian sense at all. When Boscovich investigates the 
attraction of " bodies," he introduces the idea of a cause, but then only more or less as a 
convenient phrase. Although, as a philosopher, Boscovich denies that there is any possibility 
of a fortuitous circumstance (and here indeed we may admit a prejudice derived from 
experience ; for he states that what we call fortuitous is merely something for which we, 
in our limited intelligence, can assign no cause), yet with him the existent thing is motion 
and not force. The latter word is merely a convenient phrase to describe the " product " of 
" mass " and " acceleration." 

To sum up, it would seem that the curve of Boscovich is an acceleration-interval graph ; 
and it is a mistake to refer to his cosmic system as a system of " force-centres." His material 
points have zero volume, zero mass, and exert zero force. In fact, if one material point 
alone existed outside the mind, and there were no material point forming part of the mind, 
then this single external point could in no way be perceived. In other words, a single 
point would give no sense-datum apart from another point ; and thus single points might 
be considered as not perceptible in themselves, but as becoming so in relation to other 
material points. This seems to be the logical deduction from the strict sense of the 
definition given by Boscovich ; what Boscovich himself thought is given in the supplements 
that follow the third part of the treatise. Nevertheless, the phraseology of " attraction " 
and " repulsion " is so much more convenient than that of " acceleration of the velocity of 
approach " and " acceleration of the velocity of recession," that it will be used in what 
follows : as it has been used throughout the translation of the treatise. 

There is still another point to be considered before we take up the study of the Boscovich 
curve ; namely, whether we are to consider Boscovich as, consciously or unconsciously, an 
atomist in the strict sense of the word. The practical test for this question would seem 
to be simply whether the divisibility of matter was considered to be limited or unlimited. 
Boscovich himself appears to be uncertain of his ground, hardly knowing which point of 
view is the logical outcome of his definition of a material point. For, in Art. 394, he denies 
infinite divisibility ; but he admits infinite componibility. The denial of infinite divisibility 
is necessitated by his denial of " anything infinite in Nature, or in extension, or a self- 
determined infinitely small." The admission of infinite componibility is necessitated by 
his definition of the material point ; since it has no parts, a fresh point can always be placed 
between any two points without being contiguous to either. Now, since he denies the 
existence of the infinite and the infinitely small, the attraction or repulsion between two 
points of matter (except at what he calls the limiting intervals) must be finite : hence, since 
the attractions of masses are all by observation finite, it follows that the number of points 
in a mass must be finite. To evade the difficulty thus raised, he appeals to the scale of 
integers, in which there is no infinite number : but, as he says, the scale of integers is a 
sequence of numbers increasing indefinitely, and having no last term. Thus, into any space, 
however small, there may be crowded an indefinitely great number of material points ; this 
number can be still further increased to any extent ; and yet the number of points finally 
obtained is always finite. It would, again, seem that the system of Boscovich was not a 
material system, but a system of relations ; if it were not for the fact that he asserts, in 
Art. 7, that his view is that " the Universe does not consist of vacuum interspersed amongst 
matter, but that matter is interspersed in a vacuum and floats in it." The whole question 
is still further complicated by his remark, in Art. 393, that in the continual division of a 
body, " as soon as we reach intervals less than the distance between two material points, 
further sections will cut empty intervals and not matter " ; and yet he has postulated that 
there is no minimum value to the interval between two material points. Leaving, however, 
this question of the philosophical standpoint of Boscovich to be decided by the reader, after 
a study of the supplements that follow the third part of the treatise, let us now consider the 
curve of Boscovich. 

Boscovich, from experimental data, gives to his curve, when the interval is large, a 
branch asymptotic to the axis of intervals ; it approximates to the " hyperbola " x*y c, in 
which x represents the interval between two points, and y the vis corresponding to that 
interval, which we have agreed to call an attraction, meaning thereby, not a force, but an 


acceleration of the velocity of approach. For small intervals he has as yet no knowledge 
of the quality or quantity of his ordinates. In Supplement IV, he gives some very ingenious 
arguments against forces that are attractive at very small distances and increase indefinitely, 
such as would be the case where the law of forces was represented by an inverse power of 
the interval, or even where the force varied inversely as the interval. For the inverse fourth 
or higher power, he shows that the attraction of a sphere upon a point on its surface would 
be less than the attraction of a part of itself on this point ; for the inverse third power, he con- 
siders orbital motion, which in this case is an equiangular spiral motion, and deduces that 
after a finite time the particle must be nowhere at all. Euler, considering this case, asserted 
that on approaching the centre of force the particle must be annihilated ; Boscovich, with 
more justice, argues that this law of force must be impossible. For the inverse square law, 
the limiting case of an elliptic orbit, when the transverse velocity at the end of the major 
axis is decreased indefinitely, is taken ; this leads to rectilinear motion of the particle to the 
centre of force and a return from it ; which does not agree with the otherwise proved 
oscillation through the centre of force to an equal distance on either side. 

Now it is to be observed that this supplement is quoted from his dissertation De Lege 
Firium in Natura existentium, which was published in 1755 ; also that in 1743 he had 
published a dissertation of which the full title is : De Motu Corporis attracti in centrum 
immobile viribus decrescentibus in ratione distantiarum reciproca duplicata in spatiis non 
resistentibus. Hence it is not too much to suppose that somewhere between 1741 and 1755 
he had tried to find a means of overcoming this discrepancy ; and he was thus led to suppose 
that, in the case of rectilinear motion under an inverse square law, there was a departure 
from the law on near approach to the centre of force ; that the attraction was replaced by a 
repulsion increasing indefinitely as the distance decreased ; for this obviously would lead to 
an oscillation to the centre and back, and so come into agreement with the limiting case of 
the elliptic orbit. I therefore suggest that it was this consideration that led Boscovich to 
the doctrine of Impenetrability. However, in the treatise itself, Boscovich postulates the 
axiom of Impenetrability as applying in general, and thence argues that the force at infinitely 
small distances must be repulsive and increasing indefinitely. Hence the ordinate to the 
curve near the origin must be drawn in the opposite direction to that of the ordinates for 
sensible distances, and the area under this branch of the curve must be indefinitely great. 
That is to say, the branch must be asymptotic to the axis of ordinates ; Boscovich however 
considers that this does not involve an infinite ordinate at the origin, because the interval 
between two material points is never zero ; or, vice versa, since the repulsion increases 
indefinitely for very small intervals, the velocity of relative approach, no matter how great, 
of two material points is always destroyed before actual contact ; which necessitates a finite 
interval between two material points, and the impossibility of encounter under any circum- 
stances : the interval however, since a velocity of mutual approach may be supposed to be 
of any magnitude, can have no minimum. Two points are said to be in physical contact, 
in opposition to mathematical contact, when they are so close together that this great mutual 
repulsion is sufficiently increased to prevent nearer approach. 

Since Boscovich has these two asymptotic branches, and he postulates Continuity, 
there must be a continuous curve, with a one-valued ordinate for any interval, to represent 
the " force " at all other distances ; hence the curve must cut the axis at some point in 
between, or the ordinate must become infinite. He does not lose sight of this latter possi- 
bility, but apparently discards it for certain mechanical and physical reasons. Now, it is 
known that as the degree of a curve rises, the number of curves of that degree increases very 
rapidly ; there is only one of the first degree, the conic sections of the second degree, while 
Newton had found over three-score curves with equations of the third degree, and nobody 
had tried to find all the curves of the fourth degree. Since his curve is not one of the known 
curves, Boscovich concludes that the degree of its equation is very high, even if it is not 
transcendent. But the higher the degree of a curve, the greater the number of possible 
intersections with a given straight line ; that is to say, it is highly probable that there are a 
great many intersections of the curve with the axis ; i.e., points giving zero action for 
material points situated "at the corresponding distance from one another. Lastly, since the 
ordinate is one-valued, the equation of the curve, as stated in Supplement III, must be of 
the form P-Qy = o, where P and Q are functions of x alone. Thus we have a curve winding 
about the axis for intervals that are very small and developing finally into the hyperbola of 
the third degree for sensible intervals. This final branch, however, cannot be exactly this 
hyperbola ; for, Boscovich argues, if any finite arc of the curve ever coincided exactly with 
the hyperbola of the third degree, it would be a breach of continuity if it ever departed from 
it. Hence he concludes that the inverse square law is observed approximately only, even 
at large distances. 

As stated above, the possibility of other asymptotes, parallel to the asymptote at the 


origin, is not lost sight of. The consequence of one occurring at a very small distance from 
the origin is discussed in full. Boscovich, however, takes great pains to show that all the 
phenomena discussed can be explained on the assumption of a number of points of inter- 
section of his curve with the axis, combined with different characteristics of the arcs that lie 
between these points of intersection. There is, however, one suggestion that is very 
interesting, especially in relation to recent statements of Einstein and Weyl. Suppose that 
beyond the distances of the solar system, for which the inverse square law obtains approxi- 
mately at least, the curve of forces, after touching the axis (as it may do, since it does not 
coincide exactly with the hyperbola of the third degree), goes off to infinity in the positive 
direction ; or suppose that, after cutting the axis (as again it may do, for the reason given 
above), it once more begins to wind round the axis and finally has an asymptotic attractive 
branch. Then it is evident that the universe in which we live is a self-contained cosmic 
system ; for no point within it can ever get beyond the distance of this further asymptote. 
If in addition, beyond this further asymptote, the curve had an asymptotic repulsive branch 
and went on as a sort of replica of the curve already obtained, then no point outside our 
universe could ever enter within it. Thus there is a possibility of infinite space being 
filled with a succession of cosmic systems, each of which would never interfere with any 
other ; indeed, a mind existing in any one of these universes could never perceive the 
existence of any other universe except that in which it existed. Thus space might be in 
reality infinite, and yet never could be perceived except as finite. 

The use Boscovich makes of his curve, the ingenuity of his explanations and their logic, 
the strength or weakness of his attacks on the theories of other philosophers, are left to the 
consideration of the reader of the text. It may, however, be useful to point out certain 
matters which seem more than usually interesting. Boscovich points out that no philosopher 
has attempted to prove the existence of a centre of gravity. It would appear especially that 
he is, somehow or other, aware of the mistake made by Leibniz in his early days (a mistake 
corrected by Huygens according to the statement of Leibniz), and of the use Leibniz later 
made of the principle of moments ; Boscovich has apparently considered the work of Pascal 
and others, especially Guldinus ;, it looks almost as if (again, somehow or other) he had seen 
some description of " The Method " of Archimedes. For he proceeds to define the centre 
of gravity geometrically, and to prove that there is always a centre of gravity, or rather a 
geometrical centroid ; whereas, even for a triangle, there is no centre of magnitude, with 
which Leibniz seems to have confused a centroid before his conversation with Huygens. 
This existence proof, and the deductions from it, are necessary foundations for the centro- 
baryc analysis of Leibniz. The argument is shortly as follows : Take a plane outside, say 
to the right of, all the points of all the bodies under consideration ; find the sum of all the 
distances of all the points from this plane ; divide this sum by the number of points ; draw 
a plane to the left of and parallel to the chosen plane, at a distance from it equal to the 
quotient just found. Then, observing algebraic sign, this is a plane such that the sum of 
the distances of all the points from it is zero ; i.e., the sum of the distances of all the points 
on one side of this plane is equal arithmetically to the sum of the distances of all the points on 
the other side. Find a similar plane of equal distances in another direction ; this intersects 
the first plane in a straight line. A third similar plane cuts this straight line in a point ; 
this point is the centroid ; it has the unique property that all planes through it are planes 
of equal distances. If some of the points are conglomerated to form a particle, the sum 
of the distances for each of the points is equal to the distance of the particle multiplied by 
the number of points in the particle, i.e., by the mass of the particle. Hence follows the 
theorem for the statical moment for lines and planes or other surfaces, as well as for solids 
that have weight. 

Another interesting point, in relation to recent work, is the subject-matter of Art. 230- 
236 ; where it is shown that, due solely to the mutual forces exerted on a third point by 
two points separated by a proper interval, there is a series of orbits, approximately confocal 
ellipses, in which the third point is in a state of steady motion ; these orbits are alternately 
stable and stable. If the steady motion in a stable orbit is disturbed, by a sufficiently great 
difference of the velocity being induced by the action of a fourth point passing sufficiently 
near the third point, this third point will leave its orbit and immediately take up another 
stable orbit, after some initial oscillation about it. This elegant little theorem does not 
depend in any way on the exact form of the curve of forces, so long as there are portions of the 
curve winding about the axis for very small intervals between the points. 

It is sufficient, for the next point, to draw the reader's attention to Art. 266-278, on 
collision, and to the articles which follow on the agreement between .resolution and com- 
position of forces as a working hypothesis. From what Boscovich says, it would appear that 
philosophers of his time were much perturbed over the idea that, when a force was resolved 
into two forces at a sufficiently obtuse angle, the force itself might be less than either of 


the resolutes. Boscovich points out that, in his Theory, there is no resolution, only com- 
position ; and therefore the difficulty does not arise. In this connection he adds that there 
are no signs in Nature of anything approaching the vires viva of Leibniz. 

In Art. 294 we have Boscovich's contribution to the controversy over the correct 
measure of the " quantity of motion " ; but, as there is no attempt made to follow out the 
change in either the velocity or the square of the velocity, it cannot be said to lead to any- 
thing conclusive. As a matter of fact, Boscovich uses the result to prove the non-existence 
of vires vivce. 

In Art. 298-306 we have a mechanical exposition of reflection and refraction of light. 
This comes under the section on Mechanics, because with Boscovich light is matter moving 
with a very high velocity, and therefore reflection is a case of impact, in that it depends 
upon the destruction of the whole of the perpendicular velocity upon entering the " surface " 
of a denser medium, the surface being that part of space in front of the physical surface of 
the medium in which the particles of light are near enough to the denser medium to feel the 
influence of the last repulsive asymptotic branch of the curve of forces. If this perpendicular 
velocity is not all destroyed, the particle enters the medium, and is refracted ; in which 
case, the existence of a sine law is demonstrated. It is to be noted that the " fits " of 
alternate attraction and repulsion, postulated by Newton, follow as a natural consequence 
of the winding portion of the curve of Boscovich. 

In Art. 328-346 we have a discussion of the centre of oscillation, and the centre of 
percussion is investigated as well for masses in a plane perpendicular to the axis of rotation, 
and masses lying in a straight line, where each mass is connected with the different centres. 
Boscovich deduces from his theory the theorems, amongst others, that the centres of suspen- 
sion and oscillation are interchangeable, and that the distance between them is equal to the 
distance of the centre of percussion from the axis of rotation ; he also gives a rule for finding 
the simple equivalent pendulum. The work is completed in a letter to Fr. Scherffer, which 
is appended at the end of this volume. 

In the third section, which deals with the application of the Theory to Physics, we 
naturally do not look for much that is of value. But, in Art. 505, Boscovich evidently has 
the correct notion that sound is a longitudinal vibration of the air or some other medium ; 
and he is able to give an explanation of the propagation of the disturbance purely by means 
of the mutual forces between the particles of the medium. In Art. 507 he certainly states 
that the cause of heat is a " vigorous internal motion " ; but this motion is that of the 
" particles of fire," if it is a motion ; an alternative reason is however given, namely, that it 
may be a " fermentation of a sulphurous substance with particles of light." " Cold is 
a lack of this substance, or of a motion of it." No attention will be called to this part 
of the work, beyond an expression of admiration for the great ingenuity of a large part 
of it. 

There is a metaphysical appendix on the seat of the mind, and its nature, and on the 
existence and attributes of GOD. This is followed by two short discussions of a philosophical 
nature on Space and Time. Boscovich does not look on either of these as being in themselves 
existent ; his entities are modes of existence, temporal and local. These three sections are 
full of interest for the modern philosophical reader. 

Supplement V is a theoretical proof, purely derived from the theory of mutual actions 
between points of matter, of the law of the lever ; this is well worth study. 

There are two points of historical interest beyond the study of the work of Boscovich 
that can be gathered from this volume. The first is that at this time it would appear that 
the nature of negative numbers and quantities was not yet fully understood. Boscovich, to 
make his curve more symmetrical, continues it to the left of the origin as a reflection in the 
axis of ordinates. It is obvious, however, that, if distances to the left of the origin stand for 
intervals measured in the opposite direction to the ordinary (remembering that of the two 
points under consideration one is supposed to be at the origin), then the force just the other 
side of the axis of ordinates must be repulsive ; but the repulsion is in the opposite direction 
to the ordinary way of measuring it, and therefore should appear on the curve represented 
by an ordinate of attraction. Thus, the curve of Boscovich, if completed, should have point 
symmetry about the origin, and not line symmetry about the axis of ordinates. Boscovich, 
however, avoids this difficulty, intentionally or unintentionally, when showing how the 
equation to the curve may be obtained, by taking z = x* as his variable, and P and Q as 
functions of z, in the equation P-Qy = o, referred to above. Note. In this connection 
(p. 410, Art. 25, 1. 5), Boscovich has apparently made a slip over the negative sign : as the 
intention is clear, no attempt has been made to amend the Latin. 

The second point is that Boscovich does not seem to have any idea of integrating between 
limits. He has to find the area, in Fig. I on p. 134, bounded by the axes, the curve and the 
ordinate ag ; this he does by the use of the calculus in Note (1) on p. 141. He assumes that 



the equation of the curve is x m y n = I, and obtains the integral - - xy + A, where A is the 


constant of integration. He states that, if n is greater than m, A = o, being the initial area 
at the origin. He is then faced with the necessity of making the area infinite when n = m, 
and still more infinite when n<jn. He says : " The area is infinite, when n = m, because 
this makes the divisor zero ; and thus the area becomes still more infinite if n<^m." Put 

into symbols, the argument is : Since -OT<O, >- > oo . The historically interesting 

n m o 

point about this is that it represents the persistance of an error originally made by Wallis 
in his Ariihmetica Infinitorum (it was Wallis who invented the sign oc to stand for " simple 
infinity," the value of i/o, and hence of /o). Wallis had justification for his error, if 
indeed it was an error in his case ; for his exponents were characteristics of certain infinite 
series, and he could make his own laws about these so that they suited the geometrical 
problems to which they were applied ; it was not necessary that they should obey the laws 
of inequality that were true for ordinary numbers. Boscovich's mistake is, of course, that 
of assuming that the constant is zero in every case ; and in this he is probably deceived by 

using the formula xy -f- A, instead of ^ B/( "-* l) -}- A, for the area. From the latter 

nm n m 

it is easily seen that since the initial area is zero, we must have A = o w/( "~ m) . If n is 

m n 

equal to or greater than m, the constant A is indeed zero ; but if n is less than m, the constant 
is infinite. The persistence of this error for so long a time, from 1655 to 1 75%> during which 
we have the writings of Newton, Leibniz, the Bernoullis and others on the calculus, seems 
to lend corroboration to a doubt as to whether the integral sign was properly understood as 
a summation between limits, and that this sum could be expressed as the difference of two 
values of the same function of those limits. It appears to me that this point is one of 
very great importance in the history of the development of mathematical thought. 

Some idea of how prolific Boscovich was as an author may be gathered from the catalogue 
of his writings appended at the end of this volume. This catalogue has been taken from the 
end of the original first Venetian edition, and brings the list up to the date of its publication, 
1763. It was felt to be an impossible task to make this list complete up to the time of the 
death of Boscovich ; and an incomplete continuation did not seem desirable. Mention 
must however be made of one other work of Boscovich at least ; namely, a work in five 
quarto volumes, published in 1785, under the title of Opera pertinentia ad Opticam et 

Finally, in order to bring out the versatility of the genius of Boscovich, we may mention 
just a few of his discoveries in science, which seem to call for special attention. In astro- 
nomical science, he speaks of the use of a telescope filled with liquid for the purpose of 
measuring the aberration of light ; he invented a prismatic micrometer contemporaneously 
with Rochon and Maskelyne. He gave methods for determining the orbit of a comet from 
three observations, and for the equator of the sun from three observations of a " spot " ; 
he carried out some investigations on the orbit of Uranus, and considered the rings of Saturn. 
In what was then the subsidiary science of optics, he invented a prism with a variable angle 
for measuring the refraction and dispersion of different kinds of glass ; and put forward a 
theory of achromatism for the objectives and oculars of the telescope. In mechanics and 
geodesy, he was apparently the first to solve the problem of the " body of greatest attraction " ; 
he successfully attacked the question of the earth's density ; and perfected the apparatus 
and advanced the theory of the measurement of the meridian. In mathematical theory, 
he seems to have recognized, before Lobachevski and Bolyai, the impossibility of a proof of 
Euclid's " parallel postulate " ; and considered the theory of the logarithms of negative 

J. M. C. 

N.B. The page numbers on the left-hand pages of the index are the pages of the 
original Latin Edition of 1763 ; they correspond with the clarendon numbers inserted 
throughout the Latin text of this edition. 


Attention is called to the following important corrections, omissions, and alternative renderings ; misprints 
involving a single letter or syllable only are given at the end of the volume. 

p. 27, 1. 8, for in one plane read in the same direction 

p. 47, 1. 62, literally on which ... is exerted 

p. 49, 1. 33, for just as ... is read so that . . . may be 

P- S3> 1- 9> a f ter a li ne a dd but not parts of the line itself 

p. 61, Art. 47, Alternative rendering: These instances make good the same point as water making its way through 

the pores of a sponge did for impenetrability ; 

p. 67, 1. 5, for it is allowable for me read I am disposed ; unless in the original libet is taken to be a misprint for licet 
p. 73, 1. 26, after nothing add in the strict meaning of the term 
p. 85, 1. 27, after conjunction add of the same point of space 

p. 91, 1. 25, Alternative rendering : and these properties might distinguish the points even in the view of the followers 
of Leibniz 

1. 5 from bottom, Alternative rendering : Not to speak of the actual form of the leaves present in the seed 
p. 115, 1. 25, after the left add but that the two outer elements do not touch each other 

1. 28, for two little spheres read one little sphere 
p. 117, 1. 41, for precisely read abstractly 
p. 125, 1. 29, for ignored read urged in reply 
p. 126, 1. 6 from bottom, it is -possible that acquirere is intended for acquiescere, with a corresponding change in the 


p. 129, Art. 162, marg. note, for on what they may be founded read in what it consists, 
p. 167, Art. 214, 1. 2 of marg. note, transpose by and on 

footnote, 1. I, for be at read bisect it at 
p. 199, 1. 24, for so that read just as 
p. 233, 1. 4 from bottom, for base to the angle read base to the sine of the angle 

last line, after vary insert inversely 

p. 307, 1. 5 from end, for motion, as (with fluids) takes place read motion from taking place 
p. 323, 1. 39, for the agitation will read the fluidity will 
P- 345 1- 3 2 > f or described read destroyed 
p. 357, 1. 44, for others read some, others of others 

1. 5 from end, for fire read a fiery and insert a comma before substance 






PUS, quod tibi offero, jam ab annis quinque Viennse editum, quo plausu 
exceptum sit per Europam, noveris sane, si Diaria publica perlegeris, inter 
quse si, ut omittam caetera, consulas ea, quae in Bernensi pertinent ad 
initium anni 1761 ; videbis sane quo id loco haberi debeat. Systema 
continet Naturalis Philosophise omnino novum, quod jam ab ipso Auctore 
suo vulgo Boscovichianum appellant. Id quidem in pluribus Academiis 
jam passim publice traditur, nee tantum in annuis thesibus, vel disserta- 
tionibus impressis, ac propugnatis exponitur, sed & in pluribus elementaribus libris pro 
juventute instituenda editis adhibetur, exponitur, & a pluribus habetur pro archetype. 
Verum qui omnem systematis compagem, arctissimum partium nexum mutuum, fcecun- 
ditatem summam, ac usum amplissimum ac omnem, quam late patet, Naturam ex unica 
simplici lege virium derivandam intimius velit conspicere, ac contemplari, hoc Opus 
consulat, necesse est. 

Haec omnia me permoverant jam ab initio, ut novam Operis editionem curarem : 
accedebat illud, quod Viennensia exemplaria non ita facile extra Germaniam itura videbam, 
& quidem nunc etiam in reliquis omnibus Europse partibus, utut expetita, aut nuspiam 
venalia prostant, aut vix uspiam : systema vero in Italia natum, ac ab Auctore suo pluribus 
hie apud nos jam dissertationibus adumbratum, & casu quodam Viennae, quo se ad breve 
tempus contulerat, digestum, ac editum, Italicis potissimum typis, censebam, per univer- 
sam Europam disseminandum. Et quidem editionem ipsam e Viennensi exemplari jam 
turn inchoaveram ; cum illud mihi constitit, Viennensem editionem ipsi Auctori, post cujus 
discessum suscepta ibi fuerat, summopere displicere : innumera obrepsisse typorum menda : 
esse autem multa, inprimis ea, quas Algebraicas formulas continent, admodum inordinata, 
& corrupta : ipsum eorum omnium correctionem meditari, cum nonnullis mutationibus, 
quibus Opus perpolitum redderetur magis, & vero etiam additamentis. 

Illud ergo summopere desideravi, ut exemplar acquirerem ab ipso correctum, & auctum 
ac ipsum edition! praesentem haberem, & curantem omnia per sese. At id quidem per 
hosce annos obtinere non licuit, eo universam fere Europam peragrante ; donee demum 
ex tarn longa peregrinatione redux hue nuper se contulit, & toto adstitit editionis tempore, 
ac praeter correctores nostros omnem ipse etiam in corrigendo diligentiam adhibuit ; 
quanquam is ipse haud quidem sibi ita fidit, ut nihil omnino effugisse censeat, cum ea sit 
humanas mentis conditio, ut in eadem re diu satis intente defigi non possit. 

Haec idcirco ut prima quaedam, atque originaria editio haberi debet, quam qui cum 
Viennensi contulerit, videbit sane discrimen. E minoribus mutatiunculis multae pertinent 
ad expolienda, & declaranda plura loca ; sunt tamen etiam nonnulla potissimum in pagin- 
arum fine exigua additamenta, vel mutatiunculas exiguae factae post typographicam 
constructionem idcirco tantummodo, ut lacunulae implerentur quae aliquando idcirco 
supererant, quod plures ph'ylirae a diversis compositoribus simul adornabantur, & quatuor 
simul praela sudabant; quod quidem ipso praesente fieri facile potuit, sine ulla pertur- 
batione sententiarum, & ordinis. 




! OU will be well aware, if you have read the public journals, with what applause 
the work which I now offer to you has been received throughout Europe 
since its publication at Vienna five years ago. Not to mention others, if 
you refer to the numbers of the Berne Journal for the early part of the 
year 1761, you will not fail to see how highly it has been esteemed. It 
contains an entirely new system of Natural Philosophy, which is already 
commonly known as the Boscovichian theory, from the name of its author, 
As a matter of fact, it is even now a subject of public instruction in several Universities in 
different parts ; it is expounded not only in yearly theses or dissertations, both printed & 
debated ; but also in several elementary books issued for the instruction of the young it is 
introduced, explained, & by many considered as their original. Any one, however, who 
wishes to obtain more detailed insight into the whole structure of the theory, the close 
relation that its several parts bear to one another, or its great fertility & wide scope for 
the purpose of deriving the whole of Nature, in her widest range, from a single simple law 
of forces ; any one who wishes to make a deeper study of it must perforce study the work 
here offered. 

All these considerations had from the first moved me to undertake a new edition of 
the work ; in addition, there was the fact that I perceived that it would be a matter of some 
difficulty for copies of the Vienna edition to pass beyond the confines of Germany indeed, 
at the present time, no matter how diligently they are inquired for, they are to be found 
on sale nowhere, or scarcely anywhere, in the rest of Europe. The system had its birth in 
Italy, & its outlines had already been sketched by the author in several dissertations pub- 
lished here in our own land ; though, as luck would have it, the system itself was finally 
put into shape and published at Vienna, whither he had gone for a short time. I therefore 
thought it right that it should be disseminated throughout the whole of Europe, & that 
preferably as the product of an Italian press. I had in fact already commenced an edition 
founded on a copy of the Vienna edition, when it came to my knowledge that the author 
was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ; 
that innumerable printer's errors had crept in ; that many passages, especially those that 
contain Algebraical formulae, were ill-arranged and erroneous ; lastly, that the author 
himself had in mind a complete revision, including certain alterations, to give a better 
finish to the work, together with certain additional matter. 

That being the case, I was greatly desirous of obtaining a copy, revised & enlarged 
by himself ; I also wanted to have him at hand whilst the edition was in progress, & that 
he should superintend the whole thing for himself. This, however, I was unable to procure 
during the last few years, in which he has been travelling through nearly the whole of 
Europe ; until at last he came here, a little while ago, as he returned home from his lengthy 
wanderings, & stayed here to assist me during the whole time that the edition was in 
hand. He, in addition to our regular proof-readers, himself also used every care in cor- 
recting the proof ; even then, however, he has not sufficient confidence in himself as to 
imagine that not the slightest thing has escaped him. For it is a characteristic of the human 
mind that it cannot concentrate long on the same subject with sufficient attention. 

It follows that this ought to be considered in some measure as a first & original 
edition ; any one who compares it with that issued at Vienna will soon see the difference 
between them. Many of the minor alterations are made for the purpose of rendering 
certain passages more elegant & clear ; there are, however, especially at the foot of a 
page, slight additions also, or slight changes made after the type was set up, merely for 
the purpose of filling up gaps that were left here & there these gaps being due to the 
fact that several sheets were being set at the same time by different compositors, and four 
presses were kept hard at work together. As he was at hand, this could easily be done 
without causing any disturbance of the sentences or the pagination. 


Inter mutationes occurret ordo numerorum mutatus in paragraphis : nam numerus 82 
de novo accessit totus : deinde is, qui fuerat 261 discerptus est in 5 ; demum in Appendice 
post num. 534 factse sunt & mutatiunculse nonnullae, & additamenta plura in iis, quse 
pertinent ad sedem animse. 

Supplementorum ordo mutatus est itidem ; quse enim fuerant 3, & 4, jam sunt I, & 
2 : nam eorum usus in ipso Opere ante alia occurrit. Illi autem, quod prius fuerat primum, 
nunc autem est tertium, accessit in fine Scholium tertium, quod pluribus numeris complec- 
titur dissertatiunculam integram de argumento, quod ante aliquot annos in Parisiensi 
Academia controversise occasionem exhibuit in Encyclopedico etiam dictionario attactum, 
in qua dissertatiuncula demonstrat Auctor non esse, cur ad vim exprimendam potentia 
quaepiam distantiae adhibeatur potius, quam functio. 

Accesserunt per totum Opus notulae marginales, in quibus eorum, quae pertractantur 
argumenta exponuntur brevissima, quorum ope unico obtutu videri possint omnia, & in 
memoriam facile revocari. 

Postremo loco ad calcem Operis additus est fusior catalogus eorum omnium, quse hue 
usque ab ipso Auctore sunt edita, quorum collectionem omnem expolitam, & correctam, 
ac eorum, quse nondum absoluta sunt, continuationem meditatur, aggressurus illico post 
suum regressum in Urbem Romam, quo properat. Hie catalogus impressus fuit Venetisis 
ante hosce duos annos in reimpressione ejus poematis de Solis ac Lunae defectibus. 
Porro earn, omnium suorum Operum Collectionem, ubi ipse adornaverit, typis ego meis 
excudendam suscipiam, quam magnificentissime potero. 

Haec erant, quae te monendum censui ; tu laboribus nostris fruere, & vive felix. 


Among the more Important alterations will be found a change in the order of numbering 
the paragraphs. Thus, Art. 82 is additional matter that is entirely new ; that which was 
formerly Art. 261 is now broken up into five parts ; &, in the Appendix, following Art. 
534, both some slight changes and also several additions have been made in the passages 
that relate to the Seat of the Soul. 

The order of the Supplements has been altered also : those that were formerly num- 
bered III and IV are now I and II respectively. This was done because they are required 
for use in this work before the others. To that which was formerly numbered I, but is 
now III, there has been added a third scholium, consisting of several articles that between 
them give a short but complete dissertation on that point which, several years ago caused 
a controversy in the University of Paris, the same point being also discussed in the 
Dictionnaire Encydopedique. In this dissertation the author shows that there is no reason 
why any one power of the distance should be employed to express the force, in preference 
to a function. 

Short marginal summaries have been inserted throughout the work, in which the 
arguments dealt with are given in brief ; by the help of these, the whole matter may be 
taken in at a glance and recalled to mind with ease. 

Lastly, at the end of the work, a somewhat full catalogue of the whole of the author's 
publications up to the present time has been added. Of these publications the author 
intends to make a full collection, revised and corrected, together with a continuation of 
those that are not yet finished ; this he proposes to do after his return to Rome, for which 
city he is preparing to set out. This catalogue was printed in Venice a couple of years ago 
in connection with a reprint of his essay in verse on the eclipses of the Sun and Moon. 
Later, when his revision of them is complete, I propose to undertake the printing of this 
complete collection of his works from my own type, with all the sumptuousness at my 

Such were the matters that I thought ought to be brought to your notice. May you 
enjoy the fruit of our labours, & live in happiness. 







IA.BIS veniam, Princeps Celsissime, si forte inter assiduas sacri regirninis curas 
importunus interpellator advenio, & libellum Tibi offero mole tenuem, nee 
arcana Religionis mysteria, quam in isto tanto constitutus fastigio adminis- 
tras, sed Naturalis Philosophise principia continentem. Novi ego quidem, 
quam totus in eo sis, ut, quam geris, personam sustineas, ac vigilantissimi 
sacrorum Antistitis partes agas. Videt utique Imperialis haec Aula, videt 
universa Regalis Urbs, & ingenti admiratione defixa obstupescit, qua dili- 
gentia, quo labore tanti Sacerdotii munus obire pergas. Vetus nimirum illud celeberrimum 
age, quod agis, quod ab ipsa Tibi juventute, cum primum, ut Te Romas dantem operam 
studiis cognoscerem, mihi fors obtigit, altissime jam insederat animo, id in omni 
reliquo amplissimorum munerum Tibi commissorum cursu haesit firmissime, atque idipsum 
inprimis adjectum tarn multis & dotibus, quas a Natura uberrime congestas habes, & 
virtutibus, quas tute diuturna Tibi exercitatione, atque assiduo labore comparasti, sanc- 
tissime observatum inter tarn varias forenses, Aulicas, Sacerdotales occupationes, istos Tibi 
tarn celeres dignitatum gradus quodammodo veluti coacervavit, & omnium una tarn 
populorum, quam Principum admirationem excitavit ubique, conciliavit amorem ; unde 
illud est factum, ut ab aliis alia Te, sublimiora semper, atque honorificentiora munera 
quodammodo velut avulsum, atque abstractum rapuerint. Dum Romse in celeberrimo illo, 
quod Auditorum Rotae appellant, collegio toti Christiano orbi jus diceres, accesserat 
Hetrusca Imperialis Legatio apud Romanum Pontificem exercenda ; cum repente Mech- 
liniensi Archiepiscopo in amplissima ilia administranda Ecclesia Adjutor datus, & destinatus 
Successor, possessione prsestantissimi muneris vixdum capta, ad Hispanicum Regem ab 
Augustissima Romanorum Imperatrice ad gravissima tractanda negotia Legatus es missus, 
in quibus cum summa utriusque Aulae approbatione versatum per annos quinque ditissima 
Vacciensis Ecclesia adepta est ; atque ibi dum post tantos Aularum strepitus ea, qua 
Christianum Antistitem decet, & animi moderatione, & demissione quadam, atque in omne 
hominum genus charitate, & singular! cura, ac diligentia Religionem administras, & sacrorum 
exceres curam ; non ea tantum urbs, atque ditio, sed universum Hungariae Regnum, 
quanquam exterum hominem, non ut civem suum tantummodo, sed ut Parentem aman- 
tissimum habuit, quern adhuc ereptum sibi dolet, & angitur ; dum scilicet minore, quam 
unius anni intervallo ab Ipsa Augustissima Imperatrice ad Regalem hanc Urbem, tot 
Imperatorum sedem, ac Austriacae Dominationis caput, dignum tantis dotibus explicandis 
theatrum, eocatum videt, atque in hac Celsissima Archiepiscopali Sede, accedente Romani 
Pontificis Auctoritate collocatum ; in qua Tu quidem personam itidem, quam agis, diligen- 
tissime sustinens, totus es in gravissimis Sacerdotii Tui expediendis negotiis, in iis omnibus, 
quae ad sacra pertinent, curandis vel per Te ipsum usque adeo, ut saepe, raro admodum per 






OU will pardon me, Most Noble Prince, if perchance I come to disturb at an 
inopportune moment the unremitting cares of your Holy Office, & offer 
you a volume so inconsiderable in size ; one too that contains none of the 
inner mysteries of Religion, such as you administer from the highly exalted 
position to which you are ordained ; one that merely deals with the prin- 
ciples of Natural Philosophy. I know full well how entirely your time is 
taken up with sustaining the reputation that you bear, & in performing 
the duties of a highly conscientious Prelate. This Imperial Court sees, nay, the whole of 
this Royal City sees, with what care, what toil, you exert yourself to carry out the duties of 
so great a sacred office, & stands wrapt with an overwhelming admiration. Of a truth, 
that well-known old saying, " What you do, DO," which from your earliest youth, when 
chance first allowed me to make your acquaintance while you were studying in Rome, had 
already fixed itself deeply in your mind, has remained firmly implanted there during the 
whole of the remainder of a career in which duties of the highest importance have been 
committed to your care. Your strict observance of this maxim in particular, joined with 
those numerous talents so lavishly showered upon you by Nature, & those virtues which 
you have acquired for yourself by daily practice & unremitting toil, throughout your 
whole career, forensic, courtly, & sacerdotal, has so to speak heaped upon your shoulders 
those unusually rapid advances in dignity that have been your lot. It has aroused the 
admiration of all, both peoples & princes alike, in every land ; & at the same time it has 
earned for you their deep affection. The consequence was that one office after another, 
each ever more exalted & honourable than the preceding, has in a sense seized upon you 
& borne you away a captive. Whilst you were in Rome, giving judicial decisions to the 
whole Christian world in that famous College, the Rota of Auditors, there was added the 
duty of acting on the Tuscan Imperial Legation at the Court of the Roman Pontiff. Sud- 
denly you were appointed coadjutor to the Archbishop of Malines in the administration of 
that great church, & his future successor. Hardly had you entered upon the duties of 
that most distinguished appointment, than you were despatched by the August Empress of 
the Romans as Legate on a mission of the greatest importance. You occupied yourself on 
this mission for the space of five years, to the entire approbation of both Courts, & then 
the wealthy church of Vacz obtained your services. Whilst there, the great distractions of 
a life at Court being left behind, you administer the offices of religion & discharge the 
sacred rights with that moderation of spirit & humility that befits a Christian prelate, in 
charity towards the whole race of mankind, with a singularly attentive care. So that not 
only that city & the district in its see, but the whole realm of Hungary as well, has looked 
upon you, though of foreign race, as one of her own citizens ; nay, rather as a well beloved 
father, whom she still mourns & sorrows for, now that you have been taken from her. 
For, after less than a year had passed, she sees you recalled by the August Empress herself to 
this Imperial City, the seat of a long line of Emperors, & the capital of the Dominions of 
Austria, a worthy stage for the display of your great talents ; she sees you appointed, under 
the auspices of the authority of the Roman Pontiff, to this exalted Archiepiscopal see. 
Here too, sustaining with the utmost diligence the part you play so well, you throw your- 
self heart and soul into the business of discharging the weighty duties of your priesthood, 
or in attending to all those things that deal with the sacred rites with your own hands : so 
much so that we often see you officiating, & even administering the Sacraments, in our 


haec nostra tempora exemplo, & publico operatum, ac ipsa etiam Sacramenta administrantem 
videamus in templis, & Tua ipsius voce populos, e superiore loco docentum audiamus, atque 
ad omne virtutum genus inflammantem. 

Novi ego quidem haec omnia ; novi hanc indolem, hanc animi constitutionem ; nee 
sum tamen inde absterritus, ne, inter gravissimas istas Tuas Sacerdotales curas, Philosophicas 
hasce meditationes meas, Tibi sisterem, ac tantulae libellum molis homini ad tantum culmen 
evecto porrigerem, ac Tuo vellem Nomine insignitum. Quod enim ad primum pertinet 
caput, non Theologicas tantum, sed Philosophicas etiam perquisitiones Christiano Antistite 
ego quidem dignissimas esse censeo, & universam Naturae contemplationem omnino 
arbitror cum Sacerdotii sanctitate penitus consentire. Mirum enim, quam belle ab ipsa 
consideratione Naturae ad caslestium rerum contemplationem disponitur animus, & ad 
ipsum Divinum tantae molis Conditorem assurgit, infinitam ejus Potentiam Sapientiam, 
Providentiam admiratus, quae erumpunt undique, & utique se produnt. 

Est autem & illud, quod ad supremi sacrorum Moderatoris curam pertinet providere, 
ne in prima ingenuae juventutis institutione, quae semper a naturalibus studiis exordium 
ducit, prava teneris mentibus irrepant, ac perniciosa principia, quae sensim Religionem 
corrumpant, & vero etiam evertant penitus, ac eruant a fundamentis ; quod quidem jam 
dudum tristi quodam Europae fato passim evenire cernimus, gliscente in dies malo, ut fucatis 
quibusdam, profecto perniciosissimis, imbuti principiis juvenes, turn demum sibi sapere 
videantur, cum & omnem animo religionem, & Deum ipsum sapientissimum Mundi 
Fabricatorem, atque Moderatorem sibi mente excusserint. Quamobrem qui veluti ad 
tribunal tanti Sacerdotum Principis Universae Physicae Theoriam, & novam potissimum 
Theoriam sistat, rem is quidem praestet sequissimam, nee alienum quidpiam ab ejus munere 
Sacerdotali offerat, sed cum eodem apprime consentiens. 

Nee vero exigua libelli moles deterrere me debuit, ne cum eo ad tantum Principem 
accederem. Est ille quidem satis tenuis libellus, at non & tenuem quoque rem continet. 
Argumentum pertractat sublime admodum, & nobile, in quo illustrando omnem ego quidem 
industriam coUocavi, ubi si quid praestitero, si minus infiliclter me gessero, nemo sane me 
impudentiae arguat, quasi vilem aliquam, & tanto indignam fastigio rem offeram. Habetur 
in eo novum quoddam Universae Naturalis Philosophiae genus a receptis hue usque, usi- 
tatisque plurimam discrepans, quanquam etiam ex iis, quae maxime omnium per haec tempora 
celebrantur, casu quodam praecipua quasque mirum sane in modum compacta, atque inter 
se veluti coagmentata conjunguntur ibidem, uti sunt simplicia atque inextensa Leibnitian- 
orum elementa, cum Newtoni viribus inducentibus in aliis distantiis accessum mutuum, in 
aliis mutuum recessum, quas vulgo attractiones, & repulsiones appellant : casu, inquam : 
neque enim ego conciliandi studio hinc, & inde decerpsi quaedam ad arbitrium selecta, quae 
utcumque inter se componerem, atque compaginarem : sed omni praejudicio seposito, a 
principiis exorsus inconcussis, & vero etiam receptis communiter, legitima ratiocinatione 
usus, & continue conclusionum nexu deveni ad legem virium in Natura existentium unicam, 
simplicem , continuam, quae mihi & constitutionem elementorum materiae, & Mechanicae 
leges, & generales materiae ipsius proprietates, & praecipua corporum discrimina, sua 
applicatione ita exhibuit, ut eadem in iis omnibus ubique se prodat uniformis agendi ratio, 
non ex arbitrariis hypothesibus, & fictitiis commentationibus, sed ex sola continua ratio- 
cinatione deducta. Ejusmodi autem est omnis, ut eas ubique vel definiat, vel adumbret 
combinationes elementorum, quae ad diversa prasstanda phaenomena sunt adhibendas, ad 
quas combinationes Conditoris Supremi consilium, & immensa Mentis Divinae vis ubique 
requiritur, quae infinites casus perspiciat, & ad rem aptissimos seligat, ac in Naturam 

Id mihi quidem argumentum est operis, in quo Theoriam meam expono, comprobo, 
vindico : turn ad Mechanicam primum, deinde ad Physicam applico, & uberrimos usus 
expono, ubi brevi quidem libello, sed admodum diuturnas annorum jam tredecim medita- 
tiones complector meas, eo plerumque tantummodo rem deducens, ubi demum cum 


churches (a somewhat unusual thing at the present time), and also hear you with your own 
voice exhorting the people from your episcopal throne, & inciting them to virtue of 
every kind. 

I am well aware of all this ; I know full well the extent of your genius, & your con- 
stitution of mind ; & yet I am not afraid on that account of putting into your hands, 
amongst all those weighty duties of your priestly office, these philosophical meditations of 
mine ; nor of offering a volume so inconsiderable in bulk to one who has attained to such 
heights of eminence ; nor of desiring that it should bear the hall-mark of your name. With 
regard to the first of these heads, I think that not only theological but also philosophical 
investigations are quite suitable matters for consideration by a Christian prelate ; & in 
my opinion, a contemplation of all the works of Nature is in complete accord with the 
sanctity of the priesthood. For it is marvellous how exceedingly prone the mind becomes 
to pass from a contemplation of Nature herself to the contemplation of celestial, things, & 
to give honour to the Divine Founder of such a mighty structure, lost in astonishment at 
His infinite Power & Wisdom & Providence, which break forth & disclose themselves 
in all directions & in all things. 

There is also this further point, that it is part of the duty of a religious superior to take 
care that, in the earliest training of ingenuous youth, which always takes its start from the 
study of the wonders of Nature, improper ideas do not insinuate themselves into tender 
minds ; or such pernicious principles as may gradually corrupt the belief in things Divine, 
nay, even destroy it altogether, & uproot it from its very foundations. This is what we 
have seen for a long time taking place, by some unhappy decree of adverse fate, all over 
Europe ; and, as the canker spreads at an ever increasing rate, young men, who have been 
made to imbibe principles that counterfeit the truth but are actually most pernicious doc- 
trines, do not think that they have attained to wisdom until they have banished from their 
minds all thoughts of religion and of God, the All- wise Founder and Supreme Head of the 
Universe. Hence, one who so to speak sets before the judgment-seat of such a prince of 
the priesthood as yourself a theory of general Physical Science, & more especially one that 
is new, is doing nothing but what is absolutely correct. Nor would he be offering him 
anything inconsistent with his priestly office, but on the contrary one that is in complete 
harmony with it. 

Nor, secondly, should the inconsiderable size of my little book deter me from approach- 
ing with it so great a prince. It is true that the volume of the book is not very great, but 
the matter that it contains is not unimportant as well. The theory it develops is a strik- 
ingly sublime and noble idea ; & I have done my very best to explain it properly. If in 
this I have somewhat succeeded, if I have not failed altogether, let no one accuse me of 
presumption, as if I were offering some worthless thing, something unworthy of such dis- 
tinguished honour. In it is contained a new kind of Universal Natural Philosophy, one that 
differs widely from any that are generally accepted & practised at the present time ; 
although it so happens that the principal points of all the most distinguished theories of the 
present day, interlocking and as it were cemented together in a truly marvellous way, are 
combined in it ; so too are the simple unextended elements of the followers of Leibniz, 
as well as the Newtonian forces producing mutual approach at 'some distances & mutual 
separation at others, usually called attractions and repulsions. I use the words " it so 
happens " because I have not, in eagerness to make the whole consistent, selected one thing 
here and another there, just as it suited me for the purpose of making them agree & form 
a connected whole. On the contrary, I put on one side all prejudice, & started from 
fundamental principles that are incontestable, & indeed are those commonly accepted ; I 
used perfectly sound arguments, & by a continuous chain of deduction I arrived at a 
single, simple, continuous law for the forces that exist in Nature. The application of this 
law explained to me the constitution of the elements of matter, the laws of Mechanics, the 
general properties of matter itself, & the chief characteristics of bodies, in such a manner 
that the same uniform method of action in all things disclosed itself at all points ; being 
deduced, not from arbitrary hypotheses, and fictitibus explanations, but from a single con- 
tinuous chain of reasoning. Moreover it is in all its parts of such a kind as defines, or 
suggests, in every case, the combinations of the elements that must be employed to produce 
different phenomena. For these combinations the wisdom of the Supreme Founder of the 
Universe, & the mighty power of a Divine Mind are absolutely necessary ; naught but 
one that could survey the countless cases, select those most suitable for the purpose, and 
introduce them into the scheme of Nature. 

This then is the argument of my work, in which I explain, prove & defend my theory ; 
then I apply it, in the first instance to Mechanics, & afterwards to Physics, & set forth 
the many advantages to be derived from it. Here, although the book is but small, I yet 
include the well-nigh daily meditations of the last thirteen years, carrying on my conclu- 


communibus Philosophorum consentio placitis, & ubi ea, quae habemus jam pro compertis, 
ex meis etiam deductionibus sponte fluunt, quod usque adeo voluminis molem contraxit. 
Dederam ego quidem dispersa dissertatiunculis variis Theorise meae qusedam velut specimina, 
quae inde & in Italia Professores publicos nonnullos adstipulatores est nacta, & jam ad 
exteras quoque gentes pervasit ; sed ea nunc primum tota in unum compacta, & vero etiam 
plusquam duplo aucta, prodit in publicum, quern laborem postremo hoc mense, molestiori- 
bus negotiis, quae me Viennam adduxerant, & curis omnibus exsolutus suscepi, dum in 
Italiam rediturus opportunam itineri tempus inter assiduas nives opperior, sed omnem in 
eodem adornando, & ad communem mediocrum etiam Philosophorum captum accommo- 
dando diligentiam adhibui. 

Inde vero jam facile intelliges, cur ipsum laborem meum ad Te deferre, & Tuo 
nuncupare Nomini non dubitaverim. Ratio ex iis, quae proposui, est duplex : primo quidem 
ipsum argumenti genus, quod Christianum Antistitem non modo non dedecet, sed etiam 
apprime decet : turn ipsius argumenti vis, atque dignitas, quae nimirum confirmat, & erigit 
nimium fortasse impares, sed quantum fieri per me potuit, intentos conatus meos ; nam 
quidquid eo in genere meditando assequi possum, totum ibidem adhibui, ut idcirco nihil 
arbitrer a mea tenuitate proferri posse te minus indignum, cui ut aliquem offerrem laborum 
meorum fructum quantumcunque, exposcebat sane, ac ingenti clamore quodam efnagitabat 
tanta erga me humanitas Tua, qua jam olim immerentem complexus Romae, hie etiam 
fovere pergis, nee in tanto dedignatus fastigio, omni benevolentiae significatione prosequeris. 
Accedit autem & illud, quod in hisce terris vix adhuc nota, vel etiam ignota penitus Theoria 
mea Patrocinio indiget, quod, si Tuo Nomine insignata prodeat in publicum, obtinebit sane 
validissimum, & secura vagabitur : Tu enim illam, parente velut hie orbatam suo, in dies 
nimirum discessuro, & quodammodo veluti posthumam post ipsum ejus discessum typis 
impressam, & in publicum prodeuntem tueberis, fovebisque. 

Haec sunt, quae meum Tibi consilium probent, Princeps Celsissime : Tu, qua soles 
humanitate auctorem excipere, opus excipe, & si forte adhuc consilium ipsum Tibi visum 
fuerit improbandum ; animum saltern aequus respice obsequentissimum Tibi, ac devinct- 
issimum. Vale. 

Dabam Viennce in Collegia Academico Soc. JESU 
Idibus Febr. MDCCLFIIL 


sions for the most part only up to the point where I finally agreed with the opinions com- 
monly held amongst philosophers, or where theories, now accepted as established, are the 
natural results of my deductions also ; & this has in some measure helped to diminish the 
size of the volume. I had already published some instances, so to speak, of my general 
theory in several short dissertations issued at odd times ; & on that account the theory 
has found some supporters amongst the university professors in Italy, & has already made 
its way into foreign countries. But now for the first time is it published as a whole in a 
single volume, the matter being indeed more than doubled in amount. This work I have 
carried out during the last month, being quit of the troublesome business that brought me 
to Vienna, and of all other cares ; whilst I wait for seasonable time for my return journey 
through the everlasting snow to Italy. I have however used my utmost endeavours in 
preparing it, and adapting it to the ordinary intelligence of philosophers of only moderate 

From this you will readily understand why I have not hesitated to bestow this book 
of mine upon you, & to dedicate it to you. My reason, as can be seen from what I have 
said, was twofold ; in the first place, the nature of my theme is one that is not only not 
unsuitable, but is suitable in a high degree, for the consideration of a Christian priest ; 
secondly, the power & dignity of the theme itself, which doubtless gives strength & 
vigour to my efforts perchance rather feeble, but, as far as in me lay, earnest. What- 
ever in that respect I could gain by the exercise of thought, I have applied the whole of it 
to this matter ; & consequently I think that nothing less unworthy of you can be pro- 
duced by my poor ability ; & that I should offer to you some such fruit of my labours 
was surely required of me, & as it were clamorously demanded by your great kindness 
to me ; long ago in Rome you had enfolded my unworthy self in it, & here now you 
continue to be my patron, & do not disdain, from your exalted position, to honour me 
with every mark of your goodwill. There is still a further consideration, namely, that my 
Theory is as yet almost, if not quite, unknown in these parts, & therefore needs a patron's 
support ; & this it will obtain most effectually, & will go on its way in security if it 
comes before the public franked with your name. For you will protect & cherish it, 
on its publication here, bereaved as it were of that parent whose departure in truth draws 
nearer every day ; nay rather posthumous, since it will be seen in print only after he has 

Such are my grounds for hoping that you will approve my idea, most High Prince. 
I beg you to receive the work with the same kindness as you used to show to its author ; 
&, if perchance the idea itself should fail to meet with your approval, at least regard 
favourably the intentions of your most humble & devoted servant. Farewell. 

University College of the Society of Jesus, 

February i$th, 1758. 



amice Lector, Philosophic Naturalis Theoriam ex unica lege virium 
deductam, quam & ubi jam olim adumbraverim, vel etiam ex parte explica- 
verim, y qua occasione nunc uberius pertractandum, atque augendam etiam, 
susceperim, invenies in ipso -primes partis exordia. Libuit autem hoc opus 
dividere in partes tres, quarum prima continet explicationem Theories ipsius, 
ac ejus analyticam deductionem, & vindicationem : secunda applicationem- 
satis uberem ad Mechanicam ; tertia applicationem ad Physicam. 

Porro illud inprimis curandum duxi, ut omnia, quam liceret, dilucide exponerentur, nee 
sublimiore Geometria, aut Calculo indigerent. Et quidem in prima, ac tertia parte non tantum 
nullcs analyticee, sed nee geometries demonstrations occurrunt, paucissimis qiiibusdam, quibus 
indigeo, rejectis in adnotatiunculas, quas in fine paginarum quarundam invenies. Queedam 
autem admodum pauca, quce majorem Algebra, & Geometries cognitionem requirebant, vel erant 
complicatiora aliquando, & alibi a me jam edita, in fine operis apposui, quce Supplementorum 
appellavi nomine, ubi W ea addidi, quce sentio de spatio, ac tempore, Theories mece consentanea, 
ac edita itidem jam alibi. In secunda parte, ubi ad Mechanicam applicatur Theoria,a geome- 
tricis, W aliquando etiam ab algebraicis demonstrationibus abstinere omnino non potui ; sed 
ece ejusmodi sunt, ut vix unquam requirant aliud, quam Euclideam Geometriam, & primas 
Trigonometries notiones maxime simplices, ac simplicem algorithmum. 

In prima quidem parte occurrunt Figures geometricce complures, quce prima fronte vide- 
buntur etiam complicate? rem ipsam intimius non perspectanti ; verum ece nihil aliud exhibent, 
nisi imaginem quandam rerum, quce ipsis oculis per ejusmodi figuras sistuntur contemplandce. 
Ejusmodi est ipsa ilia curva, quce legem virium exhibet. Invenio ego quidem inter omnia 
materice puncta vim quandam mutuam, quce a distantiis pendet, 5" mutatis distantiis mutatur 
ita, ut in aliis attractiva sit, in aliis repulsiva, sed certa quadam, y continua lege. Leges 
ejusmodi variationis binarum quantitatum a se invicem pendentium, uti Jiic sunt distantia, 
y vis, exprimi possunt vel per analyticam formulam, vel per geometricam curvam ; sed ilia 
prior expressio & multo plures cognitiones requirit ad Algebram pertinentes, & imaginationem 
non ita adjuvat, ut heec posterior, qua idcirco sum usus in ipsa prima operis parte, rejecta in 
Supplementa formula analytica, quce y curvam, & legem virium ab ilia expressam exhibeat. 

Porro hue res omnis reducitur. Habetur in recta indefinita, quce axis dicitur, punctum 
quoddam, a quo abscissa ipsius rectce segmenta referunt distantias. Curva linea protenditur 
secundum rectam ipsam, circa quam etiam serpit, y eandem in pluribus secat punctis : rectce 
a fine segmentorum erectce perpendiculariter usque ad curvam, exprimunt vires, quce majores 
sunt, vel minores, prout ejusmodi rectce sunt itidem majores, vel minores ; ac eesdem ex attrac- 
tivis migrant in repulsivis, vel vice versa, ubi illce ipsce perpendiculares rectce directionem 
mutant, curva ab alter a axis indefiniti plaga migrante ad alter am. Id quidem nullas requirit 
geometricas demonstrations, sed meram cognitionem vocum quarundam, quce vel ad prima per- 
tinent Geometries elementa, y notissimce sunt, vel ibi explicantur, ubi adhibentur. Notissima 
autem etiam est significatio vocis Asymptotus, unde & crus asymptoticum curvce appellatur ; 
dicitur nimirum recta asymptotus cruris cujuspiam curvce, cum ipsa recta in infinitum producta, 
ita ad curvilineum arcum productum itidem in infinitum semper accedit magis, ut distantia 
minuatur in infinitum, sed nusquam penitus evanescat, illis idcirco nunquam invicem con- 

Consider atio porro attenta curvce propositce in Fig. I, &rationis, qua per illam exprimitur 




EAR Reader, you have before you a Theory of Natural Philosophy deduced 
from a single law of Forces. You will find in the opening paragraphs of 
the first section a statement as to where the Theory has been already 
published in outline, & to a certain extent explained ; & also the occasion 
that led me to undertake a more detailed treatment & enlargement of it. 
For I have thought fit to divide the work into three parts ; the first of 
these contains the exposition of the Theory itself, its analytical deduction 
& its demonstration ; the second a fairly full application to Mechanics ; & the third an 
application to Physics. 

The most important point, I decided, was for me to take the greatest care that every- 
thing, as far as was possible, should be clearly explained, & that there should be no need for 
higher geometry or for the calculus. Thus, in the first part, as well as in the third, there 
are no proofs by analysis ; nor are there any by geometry, with the exception of a very few 
that are absolutely necessary, & even these you will find relegated to brief notes set at the 
foot of a page. I have also added some very few proofs, that required a knowledge of 
higher algebra & geometry, or were of a rather more complicated nature, all of which have 
been already published elsewhere, at the end of the work ; I have collected these under 
the heading Supplements ; & in them I have included my views on Space & Time, which 
are in accord with my main Theory, & also have been already published elsewhere. In 
the second part, where the Theory is applied to Mechanics, I have not been able to do 
without geometrical proofs altogether ; & even in some cases I have had to give algebraical 
proofs. But these are of such a simple kind that they scarcely ever require anything more 
than Euclidean geometry, the first and most elementary ideas of trigonometry, and easy 
analytical calculations. 

It is true that in the first part there are to be found a good many geometrical diagrams, 
which at first sight, before the text is considered more closely, will appear to be rather 
complicated. But these present nothing else but a kind of image of the subjects treated, 
which by means of these diagrams are set before the eyes for contemplation. The very 
curve that represents the law of forces is an instance of this. I find that between all points 
of matter there is a mutual force depending on the distance between them, & changing as 
this distance changes ; so that it is sometimes attractive, & sometimes repulsive, but always 
follows a definite continuous law. Laws of variation of this kind between two quantities 
depending upon one another, as distance & force do in this instance, may be represented 
either by an analytical formula or by a geometrical curve ; but the former method of 
representation requires far more knowledge of algebraical processes, & does not assist the 
imagination in the way that the latter does. Hence I have employed the latter method in 
the first part of the work, & relegated to the Supplements the analytical formula which 
represents the curve, & the law of forces which the curve exhibits. 

The whole matter reduces to this. In a straight line of indefinite length, which is 
called the axis, a fixed point is taken ; & segments of the straight line cut off from this 
point represent the distances. A curve is drawn following the general direction of this 
straight line, & winding about it, so as to cut it in several places. Then perpendiculars that 
are drawn from the ends of the segments to meet the curve represent the forces ; these 
forces are greater or less, according as such perpendiculars are greater or less ; & they pass 
from attractive forces to repulsive, and vice versa, whenever these perpendiculars change 
their direction, as the curve passes from one side of the axis of indefinite length to the other 
side of it. Now this requires no geometrical proof, but only a knowledge of certain terms, 
which either belong to the first elementary principles of "geometry, & are thoroughly well 
known, or are such as can be defined when they are used. The term Asymptote is well 
known, and from the same idea we speak of the branch of a curve as being asymptotic ; 
thus a straight line is said to be the asymptote to any branch of a curve when, if the straight 
line is indefinitely produced, it approaches nearer and nearer to the curvilinear arc which 
is also prolonged indefinitely in such manner that the distance between them becomes 
indefinitely diminished, but never altogether vanishes, so that the straight line & the curve 
never really meet. 

A careful consideration of the curve given in Fig. I, & of the way in which the relation 


nexus inter vires, y distantias, est utique admodum necessaria ad intelligendam Theoriam ipsam, 
cujus ea est prcecipua qucedam veluti clavis, sine qua omnino incassum tentarentur cetera ; sed 
y ejusmodi est, ut tironum, & sane etiam mediocrium, immo etiam longe infra mediocritatem 
collocatorum, captum non excedat, potissimum si viva accedat Professoris vox mediocriter etiam 
versati in Mechanica, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut 
ii etiam, qui Geometric penitus ignari sunt, paucorum admodum explicatione vocabulorum 
accidente, earn ipsis oculis intueantur omnino perspicuam. 

In tertia parte supponuntur utique nonnulla, quce demonstrantur in secunda ; sed ea ipsa 
sunt admodum pauca, & Us, qui geometricas demonstrationes fastidiunt, facile admodum exponi 
possunt res ipsce ita, ut penitus etiam sine ullo Geometries adjumento percipiantur, quanquam 
sine Us ipsa demonstratio baberi non poterit ; ut idcirco in eo differre debeat is, qui secundam 
partem attente legerit, & Geometriam calleat, ab eo, qui earn omittat, quod ille primus veritates 
in tertia parte adhibitis, ac ex secunda erutas, ad, explicationem Physicce, intuebitur per evi- 
dentiam ex ipsis demonstrationibus haustam, hie secundus easdem quodammodo per fidem Geo- 
metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis 
etiam homogeneis, prceditis lege virium proposita, posse per solam diversam ipsorum punctorum 
dispositionem aliam particulam per certum intervallum vel perpetuo attrahere, vel perpetuo 
repellere, vel nihil in earn agere, atque id ipsum viribus admodum diversis, y quce respectu diver- 
sarum particularum diver see sint, & diver see respectu partium diver sarum ejusdem particulce, 
ac aliam particulam alicubi etiam urgeant in latus, unde plurium phcenomenorum explicatio in 
Physica sponte fluit. 

Verum qui omnem Theories, y deductionum compagem aliquanto altius inspexerit, ac 
diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius 
progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Opticce questione 
prolatis Us, quce per vim attractivam, & vim repulsivam, mutata distantia ipsi attractive suc- 
cedentem, explicari poterant, hcec addidit : " Atque hcec quidem omnia si ita sint, jam Natura 
universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum 
ccelestium motus attractione gravitatis, quce est mutua inter corpora ilia omnia, & minores fere 
omnes particularum suarum motus alia aliqua vi attrahente, & repellente, qua est inter particulas 
illas mutua" Aliquanto autem inferius de primigeniis particulis agens sic habet : " Porro 
videntur mihi hce particulce primigenice non modo in se vim inertice habere, motusque leges passivas 
illas, quce ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis 
actuosis, qualia nimirum sunt gravitas, ff causa fermentationis, & cohcerentia corporum. Atque 
hcec quidem principia considero non ut occultas qualitates, quce ex specificis rerum formis oriri 
fingantur, sed ut universales Naturce leges, quibus res ipsce sunt formatce. Nam principia 
quidem talia revera existere ostendunt phenomena Naturce, licet ipsorum causce quce sint, 
nondum fuerit explicatum. Affirmare, singulas rerum species specificis prceditas esse qualita- 
tibus occultis, per quas eae vim certam in agenda habent, hoc utique est nihil dicere : at ex 
phcenomenis Naturce duo, vel tria derivare generalia motus principia, & deinde explicare, 
quemadmodum proprietates, & actiones rerum corporearum omnium ex istis principiis conse- 
quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum 
causce nondum essent cognitce. Quare motus principia supradicta proponere non dubito, cum 
per Naturam universam latissime pateant" 

Hcec ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus 
est eum, qui ad duo, vel tria generalia motus principia ex Naturce phcenomenis derivata pheeno- 
menorum explicationem reduxerit, & sua principia protulit, ex quibus inter se diversis eorum 
aliqua tantummodo explicari posse censuit. Quid igitur, ubi tf? ea ipsa tria, & alia prcecipua 
quceque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima 
ratiocinatione deductum ? At id -per meam unicam, & simplicem virium legemprcestari, patebit 
sane consideranti operis totius Synopsim quandam, quam hie subjicio ; sed multo magis opus 
ipsum diligentius pervolventi. 




\ OU will be well aware, if you have read the public journals, with what applause 
the work which I now offer to you has been received throughout Europe 
since its publication at Vienna five years ago. Not to mention others, if 
you refer to the numbers of the Berne Journal for the early part of the 
year 1761, you will not fail to see how highly it has been esteemed. It 
contains an entirely new system of Natural Philosophy, which is already 
commonly known as the Boscovichian theory, from the name of its author, 
As a matter of fact, it is even now a subject of public instruction in several Universities in 
different parts ; it is expounded not only in yearly theses or dissertations, both printed & 
debated ; but also in several elementary books issued for the instruction of the young it is 
introduced, explained, & by many considered as their original. Any one, however, who 
wishes to obtain more detailed insight into the whole structure of the theory, the close 
relation that its several parts bear to one another, or its great fertility & wide scope for 
the purpose of deriving the whole of Nature, in her widest range, from a single simple law 
of forces ; any one who wishes to make a deeper study of it must perforce study the work 
here offered. 

All these considerations had from the first moved me to undertake a new edition of 
the work ; in addition, there was the fact that I perceived that it would be a matter of some 
difficulty for copies of the Vienna edition to pass beyond the confines of Germany indeed, 
at the present time, no matter how diligently they are inquired for, they are to be found 
on sale nowhere, or scarcely anywhere, in the rest of Europe. The system had its birth in 
Italy, & its outlines had already been sketched by the author in several dissertations pub- 
lished here in our own land ; though, as luck would have it, the system itself was finally 
put into shape and published at Vienna, whither he had gone for a short time. I therefore 
thought it right that it should be disseminated throughout the whole of Europe, & that 
preferably as the product of an Italian press. I had in fact already commenced an edition 
founded on a copy of the Vienna edition, when it came to my knowledge that the author 
was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ; 
that innumerable printer's errors had crept in ; that many passages, especially those that 
contain Algebraical formulae, were ill-arranged and erroneous ; lastly, that the author 
himself had in mind a complete revision, including certain alterations, to give a better 
finish to the work, together with certain additional matter. 

That being the case, I was greatly desirous of obtaining a copy, revised & enlarged 
by himself ; I also wanted to have him at hand whilst the edition was in progress, & that 
he should superintend the whole thing for himself. This, however, I was unable to procure 
during the last few years, in which he has been travelling through nearly the whole of 
Europe ; until at last he came here, a little while ago, as he returned home from his lengthy 
wanderings, & stayed here to assist me during the whole time that the edition was in 
hand. He, in addition to our regular proof-readers, himself also used every care in cor- 
recting the proof ; even then, however, he has not sufficient confidence in himself as to 
imagine that not the slightest thing has escaped him. For it is a characteristic of the human 
mind that it cannot concentrate long on the same subject with sufficient attention. 

It follows that this ought to be considered in some measure as a first & original 
edition ; any one who compares it with that issued at Vienna will soon see the difference 
between them. Many of the minor alterations are made for the purpose of rendering 
certain passages more elegant & clear ; there are, however, especially at the foot of a 
page, slight additions also, or slight changes made after the type was set up, merely for 
the purpose of filling up gaps that were left here & there these gaps being due to the 
fact that several sheets were being set at the same time by different compositors, and four 
presses were kept hard at work together. As he was at hand, this could easily be done 
without causing any disturbance of the sentences or the pagination. 


nexus inter vires, & distantias, est utique admodum necessaria ad intelligendam Theoriam ipsam, 
cujus ea est prcecipua queedam veluti clavis, sine qua omnino incassum tentarentur cetera ; sed 
y ejusmodi est, ut tironum, & sane etiam mediocrium, immo etiam longe infra mediocritatem 
collocatorum, captum non excedat, potissimum si viva accedat Professoris vox mediocriter etiam 
versati in Mechanics, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut 
ii etiam, qui Geometric? penitus ignari sunt, paucorum admodum explicatione vocabulorum 
accidente, earn ipsis oculis intueantur omnino perspicuam, 

In tertia parte supponuntur utique nonnulla, que? demonstrantur in secunda ; sed ea ipsa 
sunt admodum pauca, & Us, qui geometricas demonstrationes fastidiunt, facile admodum exponi 
possunt res ipsee ita, ut penitus etiam sine ullo Geometric adjumento percipiantur, quanquam 
sine Us ipsa demonstratio haberi non poterit ; ut idcirco in eo differre debeat is, qui secundam 
partem attente legerit, y Geometriam calleat, ab eo, qui earn omittat, quod ille primus veritates 
in tertia parte adhibitis, ac ex secunda erutas, ad explicationem Physics, intuebitur per evi- 
dentiam ex ipsis demonstrationibus baustam, hie secundus easdem quodammodo per fidem Geo- 
metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis 
etiam bomogeneis, preeditis lege virium proposita, posse per solam diversam ipsorum punctorum 
dispositionem aliam particulam per cerium intervallum vel perpetuo attrahere, vel perpetuo 
repellere, vel nihil in earn agere, atque id ipsum viribus admodum diversis, y que? respectu diver- 
sarum particularum diver see sint, y diverse? respectu partium diver sarum ejusdem particulce, 
ac aliam particulam alicubi etiam urgeant in latus, unde plurium pheenomenorum explicatio in 
Physica sponte ftuit. 

Ferum qui omnem Theorie?, y deductionum compagem aliquanto altius inspexerit, ac 
diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius 
progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Opticce questione 
prolatis Us, qua per vim attractivam, y vim repulsivam, mutata distantia ipsi attractive? suc- 
cedentem, explicari poterant, he?c addidit : " Atque he?c quidem omnia si ita sint, jam Natura 
universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum 
ccelestium motus attractione gravitatis, quee est mutua inter corpora ilia omnia, y minores fere 
omnes particularum suarum motus alia aliqua vi attrabente, y repellente, quiz est inter particulas 
illas mutua." Aliquanto autem inferius de primigeniis particulis agens sic habet : " Porro 
videntur mihi he? particule? primigeniee non modo in se vim inertice habere, motusque leges passivas 
illas, que? ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis 
actuosis, qualia nimirum sunt gravitas, y causa fermentationis, y cohcerentia corporum. Atque 
heec quidem principia considero non ut occultas qualitates, que? ex specificis rerum formis oriri 
fingantur, sed ut universales Nature? leges, quibus res ipse? sunt formates. Nam principia 
quidem talia revera existere ostendunt phenomena Nature?, licet ipsorum cause? que? sint, 
nondum fuerit explicatum. Affirmare, singulas rerum species specificis preeditas esse qualita- 
tibus occultis, per quas eae vim certam in agenda habent, hoc utique est nihil dicere : at ex 
phcenomenis Nature? duo, vel tria derivare generalia motus principia, y deinde explicare, 
quemadmodum proprietates, y actiones rerum corporearum omnium ex istis principiis conse- 
quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum 
cause? nondum essent cognite?. Quare motus principia supradicta proponere non dubito, cum 
per Naturam universam latissime pateant" 

Hc?c ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus 
est eum, qui ad duo, vel tria generalia motus principia ex Nature? pheenomenis derivata phe?no- 
menorum explicationem reduxerit, y sua principia protulit, ex quibus inter se diversis eorum 
aliqua tantummodo explicari posse censuit. Quid igitur, ubi y ea ipsa tria, y alia preecipua 
quczque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima 
ratiocinatione deductum ? At id per meam unicam, y simplicem virium legempr<zstari,patebit 
sane consideranti operis totius Synopsim quandam, quam hie subjicio ; sed multo magis opus 
ipsum diligentius pervolventi. 




|JOU will be well aware, if you have read the public journals, with what applause 
the work which I now offer to you has been received throughout Europe 
since its publication at Vienna five years ago. Not to mention others, if 
you refer to the numbers of the Berne Journal for the early part of the 
year 1761, you will not fail to see how highly it has been esteemed. It 
contains an entirely new system of Natural Philosophy, which is already 
commonly known as the Boscovicbian theory, from the name of its author, 
As a matter of fact, it is even now a subject of public instruction in several Universities in 
different parts ; it is expounded not only in yearly theses or dissertations, both printed & 
debated ; but also in several elementary books issued for the instruction of the young it is 
introduced, explained, & by many considered as their original. Any one, however, who 
wishes to obtain more detailed insight into the whole structure of the theory, the close 
relation that its several parts bear to one another, or its great fertility & wide scope for 
the purpose of deriving the whole of Nature, in her widest range, from a single simple law 
of forces ; any one who wishes to make a deeper study of it must perforce study the work 
here offered. 

All these considerations had from the first moved me to undertake a new edition of 
the work ; in addition, there was the fact that I perceived that it would be a matter of some 
difficulty for copies of the Vienna edition to pass beyond the confines of Germany indeed, 
at the present time, no matter how diligently they are inquired for, they are to be found 
on sale nowhere, or scarcely anywhere, in the rest of Europe. The system had its birth in 
Italy, & its outlines had already been sketched by the author in several dissertations pub- 
lished here in our own land ; though, as luck would have it, the system itself was finally 
put into shape and published at Vienna, whither he had gone for a short time. I therefore 
thought it right that it should be disseminated throughout the whole of Europe, & that 
preferably as the product of an Italian press. I had in fact already commenced an edition 
founded on a copy of the Vienna edition, when it came to my knowledge that the author 
was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ; 
that innumerable printer's errors had crept in ; that many passages, especially those that 
contain Algebraical formulae, were ill-arranged and erroneous ; lastly, that the author 
himself had in mind a complete revision, including certain alterations, to give a better 
finish to the work, together with certain additional matter. 

That being the case, I was greatly desirous of obtaining a copy, revised & enlarged 
by himself ; I also wanted to have him at hand whilst the edition was in progress, & that 
he should superintend the whole thing for himself. This, however, I was unable to procure 
during the last few years, in which he has been travelling through nearly the whole of 
Europe ; until at last he came here, a little while ago, as he returned home from his lengthy 
wanderings, & stayed here to assist me during the whole time that the edition was in 
hand. He, in addition to our regular proof-readers, himself also used every care in cor- 
recting the proof ; even then, however, he has not sufficient confidence in himself as to 
imagine that not the slightest thing has escaped him. For it is a characteristic of the human 
mind that it cannot concentrate long on the same subject with sufficient attention. 

It follows that this ought to be considered in some measure as a first & original 
edition ; any one who compares it with that issued at Vienna will soon see the difference 
between them. Many of the minor alterations are made for the purpose of rendering 
certain passages more elegant & clear ; there are, however, especially at the foot of a 
page, slight additions also, or slight changes made after the type was set up, merely for 
the purpose of filling up gaps that were left here & there these gaps being due to the 
fact that several sheets were being set at the same time by different compositors, and four 
presses were kept hard at work together. As he was at hand, this could easily be done 
without causing any disturbance of the sentences or the pagination. 


Inter mutationes occurret ordo numerorum mutatus in paragraphis : nam numerus 82 
de novo accessit totus : deinde is, qui fuerat 261 discerptus est in 5 ; demum in Appendice 
post num. 534 factae sunt & mutatiunculae nonnullae, & additamenta plura in iis, quae 
pertinent ad sedem animse. 

Supplementorum ordo mutatus est itidem ; quae enim fuerant 3, & 4, jam sunt i, & 
2 : nam eorum usus in ipso Opere ante alia occurrit. UK autem, quod prius fuerat primum, 
nunc autem est tertium, accessit in fine Scholium tertium, quod pluribus numeris complec- 
titur dissertatiunculam integrant de argumento, quod ante aliquot annos in Parisiensi 
Academia controversiae occasionem exhibuit in Encyclopedico etiam dictionario attactum, 
in qua dissertatiuncula demonstrat Auctor non esse, cur ad vim exprimendam potentia 
quaepiam distantice adhibeatur potius, quam functio. 

Accesserunt per totum Opus notulae marginales, in quibus eorum, quae pertractantur 
argumenta exponuntur brevissima, quorum ope unico obtutu videri possint omnia, & in 
memoriam facile revocari. 

Postremo loco ad calcem Operis additus est fusior catalogus eorum omnium, quae hue 
usque ab ipso Auctore sunt edita, quorum collectionem omnem expolitam, & correctam, 
ac eorum, quse nondum absoluta sunt, continuationem meditatur, aggressurus illico post 
suum regressum in Urbem Romam, quo properat. Hie catalogus impressus fuit Venetisis 
ante hosce duos annos in reimpressione ejus poematis de Solis ac Lunae defectibus. 
Porro earn omnium suorum Operum Collectionem, ubi ipse adornaverit, typis ego meis 
excudendam suscipiam, quam magnificentissime potero. 

Haec erant, quae te monendum censui ; tu laboribus nostris fruere, & vive felix. 



Reader, you have before you a Theory of Natural Philosophy deduced 
from a single law of Forces. You will find in the opening paragraphs of 
the first section a statement as to where the Theory has been already 
published in outline, & to a certain extent explained ; & also the occasion 
that led me to undertake a more detailed treatment & enlargement of it. 
For I have thought fit to divide the work into three parts ; the first of 
these contains the exposition of the Theory itself, its analytical deduction 
& its demonstration ; the second a fairly full application to Mechanics ; & the third an 
application to Physics. 

The most important point, I decided, was for me to take the greatest care that every- 
thing, as far as was possible, should be clearly explained, & that there should be no need for 
higher geometry or for the calculus. Thus, in the first part, as well as in the third, there 
are no proofs by analysis ; nor are there any by geometry, with the exception of a very few 
that are absolutely necessary, & even these you will find relegated to brief notes set at the 
foot of a page. I have also added some very few proofs, that required a knowledge of 
higher algebra & geometry, or were of a rather more complicated nature, all of which have 
been already published elsewhere, at the end of the work ; I have collected these under 
the heading Supplements ; & in them I have included my views on Space & Time, which 
are in accord with my main Theory, & also have been already published elsewhere. In 
the second part, where the Theory is applied to Mechanics, I have not been able to do 
without geometrical proofs altogether ; & even in some cases I have had to give algebraical 
proofs. But these are of such a simple kind that they scarcely ever require anything more 
than Euclidean geometry, the first and most elementary ideas of trigonometry, and easy 
analytical calculations. 

It is true that in the first part there are to be found a good many geometrical diagrams, 
which at first sight, before the text is considered more closely, will appear to be rather 
complicated. But these present nothing else but a kind of image of the subjects treated, 
which by means of these diagrams are set before the eyes for contemplation. The very 
curve that represents the law of forces is an instance of this. I find that between all points 
of matter there is a mutual force depending on the distance between them, & changing as 
this distance changes ; so that it is sometimes attractive, & sometimes repulsive, but always 
follows a definite continuous law. Laws of variation of this kind between two quantities 
depending upon one another, as distance & force do in this instance, may be represented 
either by an analytical formula or by a geometrical curve ; but the former method of 
representation requires far more knowledge of algebraical processes, & does not assist the 
imagination in the way that the latter does. Hence I have employed the latter method in 
the first part of the work, & relegated to the Supplements the analytical formula which 
represents the curve, & the law of forces which the curve exhibits. 

The whole matter reduces to this. In a straight line of indefinite length, which is 
called the axis, a fixed point is taken ; & segments of the straight line cut off from this 
point represent the distances. A curve is drawn following the general direction of this 
straight line, & winding about it, so as to cut it in several places. Then perpendiculars that 
are drawn from the ends of the segments to meet the curve represent the forces ; these 
forces are greater or less, according as such perpendiculars are greater or less ; & they pass 
from attractive forces to repulsive, and vice versa, whenever these perpendiculars change 
their direction, as the curve passes from one side of the axis of indefinite length to the other 
side of it. Now this requires no geometrical proof, but only a knowledge of certain terms, 
which either belong to the first elementary principles of geometry, & are thoroughly well 
known, or are such as can be defined when they are used. The term Asymptote is well 
known, and from the same idea we speak of the branch of a curve as being asymptotic ; 
thus a straight line is said to be the asymptote to any branch of a curve when, if the straight 
line is indefinitely produced, it approaches nearer and nearer to the curvilinear arc which 
is also prolonged indefinitely in such manner that the distance between them becomes 
indefinitely diminished, but never altogether vanishes, so that the straight line & the curve 
never really meet. 

A careful consideration of the curve given in Fig. I, & of the way in which the relation 



nexus inter vires, & distantias, est utique ad.rn.odum necessaria ad intelligendam Theoriam ipsam, 
cujus ea est prescipua qucsdam veluti clavis, sine qua omnino incassum tentarentur cetera ; sea 
y ejusmodi est, ut tironum, & sane etiam mediocrium, immo etiam longe infra mediocritatem 
collocatorum, captum non excedat, potissimum si viva accedat Professoris vox mediocriter etiam 
versati in Mechanica, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut 
ii etiam, qui Geometries penitus ignari sunt, paucorum admodum explicatione vocabulorum 
accidente, earn ipsis oculis intueantur omnino perspicuam. 

In tertia parte supponuntur utique nonnulla, ques demonstrantur in secunda ; sed ea ipsa 
sunt admodum pauca, & Us, qui geometricas demonstrationes fastidiunt, facile admodum exponi 
possunt res ipsez ita, ut penitus etiam sine ullo Geometries adjumento percipiantur, quanquam 
sine Us ipsa demonstratio haberi non poterit ; ut idcirco in eo differre debeat is, qui secundam 
partem attente legerit, y Geometriam calleat, ab eo, qui earn omittat, quod ille primus veritates 
in tertia parte adbibitis, ac ex secunda erutas, ad explicationem Physices, intuebitur per evi- 
dentiam ex ipsis demonstrationibus haustam, hie secundus easdem quodammodo per fidem Geo- 
metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis 
etiam homogeneis, presditis lege virium proposita, posse per solam diversam ipsorum punctorum 
dispositionem aliam particulam per certum intervallum vel perpetuo attrahere, vel perpetuo 
repellere, vel nihil in earn agere, atque id ipsum viribus admodum diversis, y qua respectu diver- 
sarum particularum diver see sint, & diver see respectu partium diver sarum ejusdem particules, 
ac aliam particulam alicubi etiam urgeant in latus, unde plurium phesnomenorum explicatio in 
Physica sponte ftuit. 

Verum qui omnem Theories, y deductionum compagem aliquanto altius inspexerit, ac 
diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius 
progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Optices questione 
prolatis Us, ques per vim attractivam, & vim repulsivam, mutata distantia ipsi attractives suc- 
cedentem, explicari poterant, hesc addidit : " Atque h<sc quidem omnia si ita sint, jam Natura 
universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum 
ceslestium motus attractione gravitatis, qucs est mutua inter corpora ilia omnia, & minores fere 
omnes particularum suarum motus alia aliqua vi attrahente, y repellente, ques est inter particulas 
illas mutua." Aliquanto autem inferius de primigeniis particulis agens sic habet : " Porro 
videntur mihi hce particules primigenics non modo in se vim inerties habere, motusque leges passivas 
illas, ques ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis 
actuosis, qualia nimirum sunt gravitas, y causa fermentationis, y cohesrentia corporum. Atque 
hesc quidem principia considero non ut occultas qualitates, ques ex specificis rerum formis oriri 
fingantur, sed ut universales Natures leges, quibus res ipscs sunt formates. Nam principia 
quidem talia revera existere ostendunt phesnomena Natures, licet ipsorum causes ques sint, 
nondum fuerit explicatum. Affirmare, singulas rerum species specificis presditas esse qualita- 
tibus occultis, per quas eae vim certam in agenda habent, hoc utique est nihil dicere : at ex 
phesnomenis Natures duo, vel tria derivare generalia motus principia, y deinde explicare, 
quemadmodum proprietates, y actiones rerum corporearum omnium ex istis principiis conse- 
quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum 
causes nondum essent cognites. Quare motus principia supradicta proponere non dubito, cum 
per Naturam universam latissime pateant." 

Hcsc ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus 
est eum, qui ad duo, vel tria generalia motus principia ex Natures phesnomenis derivata phesno- 
menorum explicationem reduxerit, y sua principia protulit, ex quibus inter se diversis eorum 
aliqua tantummodo explicari posse censuit. Quid igitur, ubi y ea ipsa tria, y alia prcscipua 
quesque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima 
ratiocinatione deductum ? At id per meam unicam, y simplicem virium legem presstari, patebit 
sane consideranti operis totius Synopsim quandam, quam hie subjicio ; sed multo magis opus 
ipsum diligentius pervolventi. 


between the forces & the distances is represented by it, is absolutely necessary for the under- 
standing of the Theory itself, to which it is as it were the chief key, without which it would 
be quite useless to try to pass on to the rest. But it is of such a nature that it does not go 
beyond the capacity of beginners, not even of those of very moderate ability, or of classes 
even far below the level of mediocrity ; especially if they have the additional assistance of 
a teacher's voice, even though he is only moderately familiar with Mechanics. By his help, 
I am sure, the subject can be made clear to every one, so that those of them that are quite 
ignorant of geometry, given the explanation of but a few terms, may get a perfectly good 
idea of the subject by ocular demonstration. 

In the third part, some of the theorems that have been proved in the second part are 
certainly assumed, but there are very few such ; &, for those who do not care for geo- 
metrical proofs, the facts in question can be quite easily stated in such a manner that they 
can be completely understood without any assistance from geometry, although no real 
demonstration is possible without them. There is thus bound to be a difference between 
the reader who has gone carefully through the second part, & who is well versed in geo- 
metry, & him who omits the second part ; in that the former will regard the facts, that 
have been proved in the second part, & are now employed in the third part for the ex- 
planation of Physics, through the evidence derived from the demonstrations of these facts, 
whilst the second will credit these same facts through the mere faith that he has in geome- 
tricians. A specially good instance of this is the fact, that a particle composed of points 
quite homogeneous, subject to a law of forces as stated, may, merely by altering the arrange- 
ment of those points, either continually attract, or continually repel, or have no effect at 
all upon, another particle situated at a known distance from it ; & this too, with forces that 
differ widely, both in respect of different particles & in respect of different parts of the same 
particle ; & may even urge another particle in a direction at right angles to the line join- 
ing the two, a fact that readily gives a perfectly natural explanation of many physical 

Anyone who shall have studied somewhat closely the whole system of my Theory, & 
what I deduce from it, will see, I hope, that I have advanced in this kind of investigation 
much further than Newton himself even thought open to his desires. For he, in the last 
of his " Questions " in his Opticks, after stating the facts that could be explained by means 
of an attractive force, & a repulsive force that takes the place of the attractive force when 
the distance is altered, has added these words : " Now if all these things are as stated, then 
the whole of Nature must be exceedingly simple in design, & similar in all its parts, accom- 
plishing all the mighty motions of the heavenly bodies, as it does, by the attraction of 
gravity, which is a mutual force between any two bodies of the whole system ; and Nature 
accomplishes nearly all the smaller motions of their particles by some other force of attrac- 
tion or repulsion, which is mutual between any two of those particles." Farther on, when 
he is speaking about elementary particles, he says : " Moreover, it appears to me that these 
elementary particles not only possess an essential property of inertia, & laws of motion, 
though only passive, which are the necessary consequences of this property ; but they also 
constantly acquire motion from the influence of certain active principles such as, for 
instance, gravity, the cause of fermentation, & the cohesion of solids. I do not consider these 
principles to be certain mysterious qualities feigned as arising from characteristic forms of 
things, but as universal laws of Nature, by the influence of which these very things have 
been created. For the phenomena of Nature show that these principles do indeed exist, 
although their nature has not yet been elucidated. To assert that each & every species is 
endowed with a mysterious property characteristic to it, due to which it has a definite mode 
in action, is really equivalent to saying nothing at all. On the other hand, to derive from 
the phenomena of Nature two or three general principles, & then to explain how the pro- 
perties & actions of all corporate things follow from those principles, this would indeed be 
a mighty advance in philosophy, even if the causes of those principles had not at the time 
been discovered. For these reasons I do not hesitate in bringing forward the principles of 
motion given above, since they are clearly to be perceived throughout the whole range of 

These are the words of Newton, & therein he states his opinion that he indeed will 
have made great strides in philosophy who shall have reduced the explanation of phenomena 
to two or three general principles derived from the phenomena of Nature ; & he 
brought forward his own principles, themselves differing from one another, by which he 
thought that some only of the phenomena could be explained. What then if not only the 
three he mentions, but also other important principles, such as impenetrability & impul- 
sive force, be reduced to a single principle, deduced by a process of rigorous argument ! It 
will be quite clear that this is exactly what is done by my single simple law of forces, to 
anyone who studies a kind of synopsis of the whole work, which I add below ; but it will be 
iar more clear to him who studies the whole work with some earnestness, 




sex numeris exhibeo, quando, & qua occasione Theoriam meam 
invenerim, ac ubi hucusque de ea egerim in dissertationibus jam editis, quid 
ea commune habeat cum Leibnitiana, quid cum Newtoniana Theoria, in 
quo ab utraque discrepet, & vero etiam utrique praestet : addo, quid 
alibi promiserim pertinens ad aequilibrium, & oscillationis centrum, & 
quemadmodum iis nunc inventis, ac ex unico simplicissimo, ac elegant- 
issimo theoremate profluentibus omnino sponte, cum dissertatiunculam 

brevem meditarer, jam eo consilio rem aggressus ; repente mihi in opus integrum justse 

molis evaserit tractatio. 

7 Turn usque ad num. II expono Theoriam ipsam : materiam constantem punctis 

prorsus simplicibus, indivisibilibus, & inextensis, ac a se invicem distantibus, quae puncta 
habeant singula vim inertiae, & praeterea vim activam mutuam pendentem a distantiis, ut 
nimirum, data distantia, detur & magnitude, & directio vis ipsius, mutata autem distantia, 
mutetur vis ipsa, quae, imminuta distantia in infinitum, sit repulsiva, & quidem 
excrescens in infinitum : aucta autem distantia, minuatur, evanescat, mutetur in attrac- 
tivam crescentem primo, turn decrescentem, evanescentem, abeuntem iterum in repul- 
sivam, idque per multas vices, donee demum in majoribus distantiis abeat in attractivam 
decrescentem ad sensum in ratione reciproca duplicata distantiarum ; quern nexum virium 
cum distantiis, & vero etiam earum transitum a positivis ad negativas, sive a repulsivis ad 
attractivas, vel vice versa, oculis ipsis propono in vi, qua binae elastri cuspides conantur ad 
es invicem accedere, vel a se invicem recedere, prout sunt plus justo distractae, vel con- 

II Inde ad num. 16 ostendo, quo pacto id non sit aggregatum quoddam virium temere 

coalescentium, sed per unicam curvam continuam exponatur ope abscissarum exprimentium 
distantias, & ordinatarum exprimentium vires, cujus curvae ductum, & naturam expono, 
ac ostendo, in quo differat ab hyperbola ilia gradus tertii, quae Newtonianum gravitatem 
exprimit : ac demum ibidem & argumentum, & divisionem propono operis totius. 

1 6 Hisce expositis gradum facio ad exponendam totam illam analysim, qua ego ad ejusmodi 

Theoriam deveni, & ex qua ipsam arbitror directa, & solidissima ratiocinatione deduci 
totam. Contendo nimirum usque ad numerum 19 illud, in collisione corporum debere vel 
haberi compenetrationem, vel violari legem continuitatis, velocitate mutata per saltum, si 
cum inaequalibus velocitatibus deveniant ad immediatum contactum, quae continuitatis lex 
cum (ut evinco) debeat omnino observari, illud infero, antequam ad contactum deveniant 
corpora, debere mutari eorum velocitates per vim quandam, quae sit par extinguendse 
velocitati, vel velocitatum differentiae, cuivis utcunque magnae. 

19 A num. 19 ad 28 expendo effugium, quo ad eludendam argumenti mei vim utuntur ii, 

qui negant corpora dura, qua quidem responsione uti non possunt Newtoniani, & Corpus- 
culares generaliter, qui elementares corporum particulas assumunt prorsus duras : qui autem 
omnes utcunque parvas corporum particulas molles admittunt, vel elasticas, difficultatem 
non effugiunt, sed transferunt ad primas superficies, vel puncta, in quibus committeretur 
omnino saltus, & lex continuitatis violaretur : ibidem quendam verborum lusum evolvo, 
frustra adhibitum ad eludendam argumenti mei vim. 

* Series numerorum, quibus tractari incipiunt, quae sunt in textu, 





N the first six articles, I state the time at which I evolved my Theory, what i * 
led me to it, & where I have discussed it hitherto in essays already pub- 
lished : also what it has in common with the theories of Leibniz and 
Newton ; in what it differs from either of these, & in what it is really 
superior to them both. In addition I state what I have published else- 
where about equilibrium & the centre of oscillation ; & how, having found 
out that these matters followed quite easily from a single theorem of the 
most simple & elegant kind, I proposed to write a short essay thereon ; but when I set to 
work to deduce the matter from this principle, the discussion, quite unexpectedly to me, 
developed into a whole work of considerable magnitude. 

From this .until Art. II, I explain the Theory itself : that matter is unchangeable, 7 
and consists of points that are perfectly simple, indivisible, of no extent, & separated from 
one another ; that each of these points has a property of inertia, & in addition a mutual 
active force depending on the distance in such a way that, if the distance is given, both the 
magnitude & the direction of this force are given ; but if the distance is altered, so also is 
the force altered ; & if the distance is diminished indefinitely, the force is repulsive, & in 
fact also increases indefinitely ; whilst if the distance is increased, the force will be dimin- 
ished, vanish, be changed to an attractive force that first of all increases, then decreases, 
vanishes, is again turned into a repulsive force, & so on many times over ; until at greater 
distances it finally becomes an attractive force that decreases approximately in the inverse 
ratio of the squares of the distances. This connection between the forces & the distances, 
& their passing from positive to negative, or from repulsive to attractive, & conversely, I 
illustrate by the force with which the two ends of a spring strive to approach towards, or 
recede from, one another, according as they are pulled apart, or drawn together, by more 
than the natural amount. 

From here on to Art. 1 6 I show that it is not merely an aggregate of forces combined n 
haphazard, but that it is represented by a single continuous curve, by means of abscissse 
representing the distances & ordinates representing the forces. I expound the construction 
& nature of this curve ; & I show how it differs from the hyperbola of the third degree 
which represents Newtonian gravitation. Finally, here too I set forth the scope of the 
whole work & the nature of the parts into which it is divided. 

These statements having been made, I start to expound the whole of the analysis, by 16 
which I came upon a Theory of this kind, & from which I believe I have deduced the whole 
of it by a straightforward & perfectly rigorous chain of reasoning. I contend indeed, from 
here on until Art. 19, that, in the collision of solid bodies, either there must be compene- 
tration, or the Law of Continuity must be violated by a sudden change of velocity, if 
the bodies come into immediate contact with unequal velocities. Now since the Law of 
Continuity must (as I prove that it must) be observed in every case, I infer that, before 
the bodies reach the point of actual contact, their velocities must be altered by some force 
which is capable of destroying the velocity, or the difference of the velocities, no matter how 
great that may be. 

From Art. 19 to Art. 28 I consider the artifice, adopted for the purpose of evading the 19 
strength of my argument by those who deny the existence of hard bodies ; as a matter of 
fact this cannot be used as an argument against me by the Newtonians, or the Corpuscular- 
ians in general, for they assume that the elementary particles of solids are perfectly hard. 
Moreover, those who admit that all the particles of solids, however small they may be, are 
soft or elastic, yet do not escape the difficulty, but transfer it to prime surfaces, or points ; 
& here a sudden change would be made & the Law of Continuity violated. In the same 
connection I consider a certain verbal quibble, used in a vain attempt to foil the force of 
my reasoning. 

* These numbers are the numbers of the articles, in which the matters given in the text are first discussed. 

17 C 


28 Sequentibus num. 28 & 29 binas alias responsiones rejicio aliorum, quarum altera, ut 

mei argument! vis elidatur, affirmat quispiam, prima materiae elementa compenetrari, alter 
dicuntur materiae puncta adhuc moveri ad se invicem, ubi localiter omnino quiescunt, & 
contra primum effugium evinco impenetrabilitatem ex inductione ; contra secundum 
expono aequivocationem quandam in significatione vocis motus, cui aequivocationi totum 

30 Hinc num. 30, & 31 ostendo, in quo a Mac-Laurino dissentiam, qui considerata eadem, 

quam ego contemplatus sum, collisione corporum, conclusit, continuitatis legem violari, 
cum ego eandem illaesam esse debere ratus ad totam devenerim Theoriam meam. 

32 Hie igitur, ut meae deductionis vim exponam, in ipsam continuitatis legem inquire, ac 

a num. 32 ad 38 expono, quid ipsa sit, quid mutatio continua per gradus omnes intermedios, 
quae nimirum excludat omnem saltum ab una magnitudine ad aliam sine transitu per 

39 intermedias, ac Geometriam etiam ad explicationem rei in subsidium advoco : turn earn 
probo primum ex inductione, ac in ipsum inductionis principium inquirens usque ad num. 
44, exhibeo, unde habeatur ejusdem principii vis, ac ubi id adhiberi possit, rem ipsam 
illustrans exemplo impenetrabilitatis erutae passim per inductionem, donee demum ejus vim 

45 applicem ad legem continuitatis demonstrandam : ac sequentibus numeris casus evolvo 
quosdam binarum classium, in quibus continuitatis lex videtur laedi nee tamen laeditur. 

48 Post probationem principii continuitatis petitam ab inductione, aliam num. 48 ejus 

probationem aggredior metaphysicam quandam, ex necessitate utriusque limitis in quanti- 
tatibus realibus, vel seriebus quantitatum realium finitis, quae nimirum nee suo principio, 
nee suo fine carere possunt. Ejus rationis vim ostendo in motu locali, & in Geometria 

52 sequentibus duobus numeris : turn num. 52 expono difficultatem quandam, quas petitur 
ex eo, quod in momento temporis, in quo transitur a non esse ad esse, videatur juxta ejusmodi 
Theoriam debere simul haberi ipsum esse, & non esse, quorum alterum ad finem praecedentis 
seriei statuum pertinet, alterum ad sequentis initium, ac solutionem ipsius fuse evolvo, 
Geometria etiam ad rem oculo ipsi sistendam vocata in auxilium. 

63 Num. 63, post epilogum eorum omnium, quae de lege continuitatis sunt dicta, id 

principium applico ad excludendum saltum immediatum ab una velocitate ad aliam, sine 
transitu per intermedias, quod & inductionem laederet pro continuitate amplissimam, & 
induceret pro ipso momento temporis, in quo fieret saltus, binas velocitates, ultimam 
nimirum seriei praecedentis, & primam novas, cum tamen duas simul velocitates idem mobile 
habere omnino non possit. Id autem ut illustrem, & evincam, usque ad num. 72 considero 
velocitatem ipsam, ubi potentialem quandam, ut appello, velocitatem ab actuali secerno, 
& multa, quae ad ipsarum naturam, ac mutationes pertinent, diligenter evolvo, nonnullis 
etiam, quae inde contra meae Theoriae probationem objici possunt, dissolutis. 

His expositis conclude jam illud ex ipsa continuitate, ubi corpus quodpiam velocius 
movetur post aliud lentius, ad contactum immediatum cum ilia velocitatum inaequalitate 
deveniri non posse, in quo scilicet contactu primo mutaretur vel utriusque velocitas, vel 
alterius, per saltum, sed debere mutationem velocitatis incipere ante contactum ipsum. 

73 Hinc num. 73 infero, debere haberi mutationis causam, quae appelletur vis : turn num. 74 

74 hanc vim debere esse mutuam, & agere in partes contrarias, quod per inductionem evinco, 

75 & inde infero num. 75, appellari posse repulsivam ejusmodi vim mutuam, ac ejus legem 
exquirendam propono. In ejusmodi autem perquisitione usque ad num. 80 invenio illud, 
debere vim ipsam imminutis distantiis crescere in infinitum ita ut par sit extinguendae 
velocitati utcunque magnse ; turn & illud, imminutis in infinitum etiam distantiis, debere 
in infinitum augeri, in maximis autem debere esse e contrario attractivam, uti est gravitas : 
inde vero colligo limitem inter attractionem, & repulsionem : turn sensim plures, ac etiam 
plurimos ejusmodi limites invenio, sive transitus ab attractione ad repulsionem, & vice 
versa, ac formam totius curvae per ordinatas suas exprimentis virium legem determino. 


In the next articles, 28 & 29, I refute a further pair of arguments advanced by others ; 28 
in the first of these, in order to evade my reasoning, someone states that there is compene- 
tration of the primary elements of matter ; in the second, the points of matter are said to 
be moved with regard to one another, even when they are absolutely at rest as regards 
position. In reply to the first artifice, I prove the principle of impenetrability by induc- 
tion ; & in reply to the second, I expose an equivocation in the meaning of the term motion, 
an equivocation upon which the whole thing depends. 

Then, in Art. 30, 31, I show in what respect I differ from Maclaurin, who, having 30 
considered the same point as myself, came to the conclusion that in the collision of bodies 
the Law of Continuity was violated ; whereas I obtained the whole of my Theory from the 
assumption that this law must be unassailable. 

At this point therefore, in order that the strength of my deductive reasoning might 32 
be shown, I investigate the Law of Continuity ; and from Art. 32 to Art. 38, I set forth its 
nature, & what is meant by a continuous change through all intermediate stages, such as 
to exclude any sudden change from any one magnitude to another except by a passage 
through intermediate stages ; & I call in geometry as well to help my explanation of the 
matter. Then I investigate its truth first of all by induction ; &, investigating the prin- 39 
ciple of induction itself, as far as Art. 44, 1 show whence the force of this principle is derived, 
& where it can be used. I give by way of illustration an example in which impenetrability 
is derived entirely by induction ; & lastly I apply the force of the principle to demonstrate 
the Law of Continuity. In the articles that follow I consider certain cases of two kinds, 45 
in which the Law of Continuity appears to be violated, but is not however really violated. 

After this proof of the principle of continuity procured through induction, in Art. 48, 48 
I undertake another proof of a metaphysical kind, depending upon the necessity of a limit 
on either side for either real quantities or for a finite series of real quantities ; & indeed it 
is impossible that these limits should be lacking, either at the beginning or the end. I 
demonstrate the force of this reasoning in the case of local motion, & also in geometry, in the 
next two articles. Then in Art. 52 I explain a certain difficulty, which is derived from the S 2 
fact that, at the instant at which there is a passage from non-existence to existence, it appears 
according to a theory of this kind that we must have at the same time both existence and 
non-existence. For one of these belongs to the end of the antecedent series of states, & the 
other to the beginning of the consequent series. I consider fairly fully' the solution of this 
problem ; and I call in geometry as well to assist in giving a visual representation of the 

In Art. 63, after summing up all that has been said about the Law of Continuity, I 63 
apply the principle to exclude the possibility of any sudden change from one velocity to 
another, except by passing through intermediate velocities ; this would be contrary to the 
very full proof that I give for continuity, as it would lead to our having two velocities at 
the instant at which the change occurred. That is to say, there would be the final velocity 
of the antecedent series, & the initial velocity of the consequent series ; in spite of the fact 
that it is quite impossible for a moving body to have two different velocities at the same 
time. Moreover, in order to illustrate & prove the point, from here on to Art. 72, I 
consider velocity itself ; and I distinguish between a potential velocity, as I call it, & an 
actual velocity ; I also investigate carefully many matters that relate to the nature of these 
velocities & to their changes. Further, I settle several difficulties that can be brought 
up in opposition to the proof of my Theory, in consequence. 

This done, I then conclude from the principle of continuity that, when one body with 
a greater velocity follows after another body having a less velocity, it is impossible that 
there should ever be absolute contact with such an inequality of velocities ; that is to say, 
a case of the velocity of each, or of one or the other, of them being changed suddenly at 
the instant of contact. I assert on the other hand that the change in the velocities must 
begin before contact. Hence, in Art. 73, I infer that there must be a cause for this change : 73 
which is to be called " force." Then, in Art. 74, I prove that this force is a mutual one, & 74 
that it acts in opposite directions ; the proof is by induction. From this, in Art. 75, I 75 
infer that such a mutual force may be said to be repulsive ; & I undertake the investigation 
of the law that governs it. Carrying on this investigation as far as Art. 80, I find that this 
force must increase indefinitely as the distance is diminished, in order that it may be capable 
of destroying any velocity, however great that velocity may be. Moreover, I find that, 
whilst the force must be indefinitely increased as the distance is indefinitely decreased, it 
must be on the contrary attractive at very great distances, as is the case for gravitation. 
Hence I infer that there must be a limit-point forming a boundary between attraction & 
repulsion ; & then by degrees I find more, indeed very many more, of such limit-points, 
or points of transition from attraction to repulsion, & from repulsion to attraction ; & I 
determine the form of the entire curve, that expresses by its ordinates the law of these forces. 


8 1 Eo usque virium legem deduce, ac definio ; turn num. 81 eruo ex ipsa lege consti- 

tutionem elementorum materiae, quae debent esse simplicia, ob repulsionem in minimis 
distantiis in immensum auctam ; nam ea, si forte ipsa elementa partibus constarent, nexum 
omnem dissolveret. Usque ad num. 88 inquire in illud, an hasc elementa, ut simplicia esse 
debent, ita etiam inextensa esse debeant, ac exposita ilia, quam virtualem extensionem 
appellant, eandem exclude inductionis principio, & difficultatem evolvo turn earn, quae peti 
possit ab exemplo ejus generis extensionis, quam in anima indivisibili, & simplice per aliquam 
corporis partem divisibilem, & extensam passim admittunt : vel omnipraesentiae Dei : turn 
earn, quae peti possit ab analogia cum quiete, in qua nimirum conjungi debeat unicum 
spatii punctum cum serie continua momentorum temporis, uti in extensione virtuali unicum 
momentum temporis cum serie continua punctorum spatii conjungeretur, ubi ostendo, nee 
quietem omnimodam in Natura haberi usquam, nee adesse semper omnimodam inter 

88 tempus, & spatium analogiam. Hie autem ingentem colligo ejusmodi determinationis 
fructum, ostendens usque ad num. 91, quantum prosit simplicitas, indivisibilitas, inextensio 
elementorum materiae, ob summotum transitum a vacuo continue per saltum ad materiam 
continuam, ac ob sublatum limitem densitatis, quae in ejusmodi Theoria ut minui in 
infinitum potest, ita potest in infinitum etiam augeri, dum in communi, ubi ad contactum 
deventum est, augeri ultra densitas nequaquam potest, potissimum vero ob sublatum omne 
continuum coexistens, quo sublato & gravissimae difficultates plurimse evanescunt, & 
infinitum actu existens habetur nullum, sed in possibilibus tantummodo remanet series 
finitorum in infinitum producta. 

91 His definitis, inquire usque ad num. 99 in illud, an ejusmodi elementa sint censenda 

homogenea, an heterogenea : ac primo quidem argumentum pro homogeneitate saltern in 
eo, quod pertinet ad totam virium legem, invenio in homogenietate tanta primi cruris 
repulsivi in minimis distantiis, ex quo pendet impenetrabilitas, & postremi attractivi, quo 
gravitas exhibetur, in quibus omnis materia est penitus homogenea. Ostendo autem, nihil 
contra ejusmodi homogenietatem evinci ex principio Leibnitiano indiscernibilium, nihil ex 
inductione, & ostendo, unde tantum proveniat discrimen in compositis massulis, ut in 
frondibus, & foliis ; ac per inductionem, & analogiam demonstro, naturam nos ad homo- 
geneitatem elementorum, non ad heterogeneitatem deducere. 

100 Ea ad probationem Theoriae pertinent ; qua absoluta, antequam inde fructus colli- 
gantur multiplices, gradum hie facio ad evolvendas difficultates, quae vel objectae jam sunt, 
vel objici posse videntur mihi, primo quidem contra vires in genere, turn contra meam 
hanc expositam, comprobatamque virium legem, ac demum contra puncta ilia indivisibilia, 
& inextensa, quae ex ipsa ejusmodi virium lege deducuntur. 

101 Primo quidem, ut iis etiam faciam satis, qui inani vocabulorum quorundam sono 
perturbantur, a num. 101 ad 104 ostendo, vires hasce non esse quoddam occultarum 
qualitatum genus, sed patentem sane Mechanismum, cum & idea earum sit admodum 
distincta, & existentia, ac lex positive comprobata ; ad Mechanicam vero pertineat omnis 

104 tractatio de Motibus, qui a datis viribus etiam sine immediate impulsu oriuntur. A num. 
104 ad 106 ostendo, nullum committi saltum in transitu a repulsionibus ad attractiones, 

1 06 & vice versa, cum nimirum per omnes inter medias quantitates is transitus fiat. Inde vero 
ad objectiones gradum facio, quae totam curvas formam impetunt. Ostendo nimirum usque 
ad num. 116, non posse omnes repulsiones a minore attractione desumi ; repulsiones ejusdem 
esse seriei cum attractionibus, a quibus differant tantummodo ut minus a majore, sive ut 
negativum a positivo ; ex ipsa curvarum natura, quae, quo altioris sunt gradus, eo in 
pluribus punctis rectam secare possunt, & eo in immensum plures sunt numero ; haberi 
potius, ubi curva quaeritur, quae vires exprimat, indicium pro curva ejus naturae, ut rectam 
in plurimis punctis secet, adeoque plurimos secum afferat virium transitus a repulsivis ad 
attractivas, quam pro curva, quae nusquam axem secans attractiones solas, vel solas pro 
distantiis omnibus repulsiones exhibeat : sed vires repulsivas, & multiplicitatem transituum 
esse positive probatam, & deductam totam curvas formam, quam itidem ostendo, non esse 
ex arcubus natura diversis temere coalescentem, sed omnino simplicem, atque earn ipsam 


So far I have been occupied in deducing and settling the law of these forces. Next, 
in Art. 8r, I derive from this law the constitution of the elements of matter. These must be 81 
quite simple, on account of the repulsion at very small distances being immensely great ; 
for if by chance those elements were made up of parts, the repulsion would destroy all 
connections between them. Then, as far as Art. 88, I consider the point, as to whether 
these elements, as they must be simple, must therefore be also of no extent ; &, having ex- 
plained what is called " virtual extension," I reject it by the principle of induction. I 
then consider the difficulty which may be brought forward from an example of this kind of 
extension ; such as is generally admitted in the case of the indivisible and one-fold soul 
pervading a divisible & extended portion of the body, or in the case of the omnipresence 
of GOD. Next I consider the difficulty that may be brought forward from an analogy with 
rest ; for here in truth one point of space must be connected with a continuous series of 
instants of time, just as in virtual extension a single instant of time would be connected with 
a continuous series of points of space. I show that there can neither be perfect rest any- gg 
where in Nature, nor can there be at all times a perfect analogy between time and space. 
In this connection, I also gather a large harvest from such a conclusion as this ; showing, 
as far as Art. 91, the great advantage of simplicity, indivisibility, & non-extension in the 
elements of matter. For they do away with the idea of a passage from a continuous vacuum 
to continuous matter through a sudden change. Also they render unnecessary any limit 
to density : this, in a Theory like mine, can be just as well increased to an indefinite extent, 
as it can be indefinitely decreased : whilst in the ordinary theory, as soon as contact takes 
place, the density cannot in any way be further increased. But, most especially, they do 
away with the idea of everything continuous coexisting ; & when this is done away with, 
the majority of the greatest difficulties vanish. Further, nothing infinite is found actually 
existing ; the only thing possible that remains is a series of finite things produced inde- 

These things being settled, I investigate, as far as Art. 99, the point as to whether QJ 
elements of this kind are to be considered as being homogeneous or heterogeneous. I find 
my first evidence in favour of homogeneity at least as far as the complete law of forces 
is concerned in the equally great homogeneity of the first repulsive branch of my curve 
of forces for very small distances, upon which depends impenetrability, & of the last attrac- 
tive branch, by which gravity is represented. Moreover I show that there is nothing that 
can be proved in opposition to homogeneity such as this, that can be derived from either 
the Leibnizian principle of " indiscernibles," or by induction. I also show whence arise 
those differences, that are so great amongst small composite bodies, such as we see in boughs 
& leaves ; & I prove, by induction & analogy, that the very nature of things leads us to 
homogeneity, & not to heterogeneity, for the elements of matter. 

These matters are all connected with the proof of my Theory. Having accomplished IO o 
this, before I start to gather the manifold fruits to be derived from it, I proceed to consider 
the objections to my theory, such as either have been already raised or seem to me capable 
of being raised ; first against forces in general, secondly against the law of forces that I 
have enunciated & proved, & finally against those indivisible, non-extended points that 
are deduced from a law of forces of this kind. 

First of all then, in order that I may satisfy even those who are confused over the 101 
empty sound of certain terms, I show, in Art. 101 to 104, that these forces are not some 
sort of mysterious qualities ; but that they form a readily intelligible mechanism, since 
both the idea of them is perfectly distinct, as well as their existence, & in addition the law 
that governs them is demonstrated in a direct manner. To Mechanics belongs every dis- 
cussion concerning motions that arise from given forces without any direct impulse. In 
Art. 104 to 106, I show that no sudden change takes place in passing from repulsions to 104 
attractions or vice versa ; for this transition is made through every intermediate quantity. 
Then I pass on to consider the objections that are made against the whole form of my 106 
curve. I show indeed, from here on to Art. 116, that all repulsions cannot be taken to 
come from a decreased attraction ; that repulsions belong to the self-same series as attrac- 
tions, differing from them only as less does from more, or negative from positive. From 
the very nature of the curves (for which, the higher the degree, the more points there are 
in which they can intersect a right line, & vastly more such curves there are), I deduce 
that there is more reason for assuming a curve of the nature of mine (so that it may cut a 
right line in a large number of points, & thus give a large number of transitions of the forces 
from repulsions to attractions), than for assuming a curve that, since it does not cut 
the axis anywhere, will represent attractions alone, or repulsions alone, at all distances. 
Further, I point out that repulsive forces, and a multiplicity of transitions are directly 
demonstrated, & the whole form of the curve is a matter of deduction ; & I also show that 
it is not formed of a number of arcs differing in nature connected together haphazard ; 


simplicitatem in Supplementis cvidentissime demonstro, exhibens methodum, qua deveniri 
possit ad aequationem ejusmodi curvse simplicem, & uniformem ; licet, ut hie ostendo, ipsa 
ilia lex virium possit mente resolvi in plures, quae per plures curvas exponantur, a quibus 
tamen omnibus ilia reapse unica lex, per unicam illam continuant, & in se simplicem curvam 

121 A num. 121 refello, quae objici possunt a lege gravitatis decrescentis in ratione reciproca 

duplicata distantiarum, quae nimirum in minimis distantiis attractionem requirit crescentem 
in infinitum. Ostendo autem, ipsam non esse uspiam accurate in ejusmodi ratione, nisi 
imaginarias resolutiones exhibeamus ; nee vero ex Astronomia deduci ejusmodi legem 
prorsus accurate servatam in ipsis Planetarum, & Cometarum distantiis, sed ad summum ita 

124 proxime, ut differentia ab ea lege sit perquam exigua : ac a num. 124 expendo argumentum, 
quod pro ejusmodi lege desumi possit ex eo, quod cuipiam visa sit omnium optima, & 
idcirco electa ab Auctore Naturae, ubi ipsum Optimismi principium ad trutinam revoco, ac 
exclude, & vero illud etiam evinco, non esse, cur omnium optima ejusmodi lex censeatur : 
in Supplementis vero ostendo, ad qua; potius absurda deducet ejusmodi lex, & vero etiam 
aliae plures attractionis, quae imminutis in infinitum distantiis excrescat in infinitum. 

131 Num. 131 a viribus transeo ad elementa, & primum ostendo, cur punctorum inexten- 

sorum ideam non habeamus, quod nimirum earn haurire non possumus per sensus, quos 
solae massae, & quidem grandiores, afficiunt, atque idcirco eandem nos ipsi debemus per 
reflexionem efformare, quod quidem facile possumus. Ceterum illud ostendo, me non 
inducere primum in Physicam puncta indivisibilia, & inextensa, cum eo etiam Leibnitianae 
monades recidant, sed sublata extensione continua difficultatem auferre illam omnem, quae 
jam olim contra Zenonicos objecta, nunquam est satis soluta, qua fit, ut extensio continua 
ab inextensis effici omnino non possit. 

140 Num. 140 ostendo, inductionis principium contra ipsa nullam habere vim, ipsorum 

autem existentiam vel inde probari, quod continuitas se se ipsam destruat, & ex ea assumpta 
probetur argumentis a me institutis hoc ipsum, prima elementa esse indivisibilia, & inextensa, 

143 nee ullum haberi extensum continuum. A num. 143 ostendo, ubi continuitatem admittam, 
nimirum in solis motibus ; ac illud explico, quid mihi sit spatium, quid tempus, quorum 
naturam in Supplementis multo uberius expono. Porro continuitatem ipsam ostendo a 
natura in solis motibus obtineri accurate, in reliquis affectari quodammodo ; ubi & exempla 
quaedam evolvo continuitatis primo aspectu violatae, in quibusdam proprietatibus luminis, 
ac in aliis quibusdam casibus, in quibus quaedam crescunt per additionem partium, non (ut 
ajunt) per intussumptionem. 


153 A num. 153 ostendo, quantum haec mea puncta a spiritibus differant ; ac illud etiam 

evolvo, unde fiat, ut in ipsa idea corporis videatur includi extensio continua, ubi in ipsam 
idearum nostrarum originem inquire, & quae inde praejudicia profluant, expono. Postremo 

165 autem loco num. 165 innuo, qui fieri possit, ut puncta inextensa, & a se invicem distantia, 
in massam coalescant, quantum libet, cohaerentem, & iis proprietatibus praeditam, quas in 
corporibus experimur, quod tamen ad tertiam partem pertinet, ibi multo uberius pertrac- 
tandum ; ac ibi quidem primam hanc partem absolve. 


166 Num. 166 hujus partis argumentum propono ; sequenti vero 167, quae potissimum in 

curva virium consideranda sint, enuncio. Eorum considerationem aggressus, primo quidem 

1 68 usque ad num. 172 in ipsos arcus inquire, quorum alii attractivi, alii repulsivi, alii asym- 
ptotici, ubi casuum occurrit mira multitudo, & in quibusdam consectaria notatu digna, ut 
& illud, cum ejus formae curva plurium asymptotorum esse possit, Mundorum prorsus 
similium seriem posse oriri, quorum alter respectu alterius vices agat unius, & indissolubilis 


but that it is absolutely one-fold. This one-fold character I demonstrate in the Supple- 
ments in a very evident manner, giving a method by which a simple and uniform equation 
may be obtained for a curve of this kind. Although, as I there point out, this law of forces 
may be mentally resolved into several, and these may be represented by several correspond- 
ing curves, yet that law, actually unique, may be compounded from all of these together 
by means of the unique, continuous & one-fold curve that I give. 

In Art. 121, I start to give a refutation of those objections that may be raised from I2 i 
a consideration of the fact that the law of gravitation, decreasing in the inverse duplicate 
ratio of the distances, demands that there should be an attraction at very small distances, 
& that it should increase indefinitely. However, I show that the law is nowhere exactly in 
conformity with a ratio of this sort, unless we add explanations that are merely imaginative ; 
nor, I assert, can a law of this kind be deduced from astronomy, that is followed with per- 
fect accuracy even at the distances of the planets & the comets, but one merely that is at 
most so very nearly correct, that the difference from the law of inverse squares is very 
slight. From Art. 124 onwards,! examine the value of the argument that can be drawn 124 
in favour of a law of this sort from the view that, as some have thought, it is the best of 
all, & that on that account it was selected by the Founder of Nature. In connection with 
this I examine the principle of Optimism, & I reject it ; moreover I prove conclusively 
that there is no reason why this sort of law should be supposed to be the best of all. Fur- 
ther in the Supplements, I show to what absurdities a law of this sort is more likely to lead ; 
& the same thing for other laws of an attraction that increases indefinitely as the distance 
is diminished indefinitely. 

In Art. 131 I pass from forces to elements. I first of all show the reason why we may 1*1 
not appreciate the idea of non-extended points ; it is because we are unable to perceive 
them by means of the senses, which are only affected by masses, & these too must be of 
considerable size. Consequently we have to build up the idea by a process of reasoning ; 
& this we can do without any difficulty. In addition, I point out that I am not the first 
to introduce indivisible & non-extended points into physical science ; for the " monads " 
of Leibniz practically come to the same thing. But I show that, by rejecting the idea of 
continuous extension, I remove the whole of the difficulty, which was raised against the 
disciples of Zeno in years gone by, & has never been answered satisfactorily ; namely, the 
difficulty arising from the fact that by no possible means can continuous extension be 
made up from things of no extent. 

In Art. 140 I show that the principle of induction yields no argument against these 140 
indivisibles ; rather their existence is demonstrated by that principle, for continuity is 
self-contradictory. On this assumption it may be proved, by arguments originated by 
myself, that the primary elements are indivisible & non-extended, & that there does not 
exist anything possessing the property of continuous extension. From Art. 143 onwards, j ., 
I point out the only connection in which I shall admit continuity, & that is in motion. 
I state the idea that I have with regard to space, & also time : the nature of these I explain 
much more fully in the Supplements. Further, I show that continuity itself is really a 
property of motions only, & that in all other things it is more or less a false assumption. 
Here I also consider some examples in which continuity at first sight appears to be 
violated, such as in some of the properties of light, & in certain other cases where things 
increase by addition of parts, and not by intussumption, as it is termed. 

From Art. 153 onwards, I show how greatly these points of mine differ from object- 153 
souls. I consider how it comes about that continuous extension seems to be included 
in the very idea of a body ; & in this connection, I investigate the origin of our ideas 
& I explain the prejudgments that arise therefrom. Finally, in Art. 165, I lightly 165 
sketch what might happen to enable points that are of no extent, & at a distance from 
one another, to coalesce into a coherent mass of any size, endowed with those properties 
that we experience in bodies. This, however, belongs to the third part ; & there it will be 
much more fully developed. This finishes the first part. 


In Art. 1 66 I state the theme of this second part ; and in Art. 167 I declare what 166 
matters are to be considered more especially in connection with the curve of forces. Com- 
ing to the consideration of these matters, I first of all, as far as Art. 172, investigate the 168 
arcs of the curve, some of which are attractive, some repulsive and some asymptotic. Here 
a marvellous number of different cases present themselves, & to some of them there are 
noteworthy corollaries ; such as that, since a curve of this kind is capable of possessing a 
considerable number of asymptotes, there can arise a series of perfectly similar cosmi, each 
of which will act upon all the others as a single inviolate elementary system. From Art. 172 


172 element!. Ad. num. 179 areas contemplor arcubus clausas, quae respondentes segmento axis 
cuicunque, esse possunt magnitudine utcunque magnae, vel parvae, sunt autem mensura 

179 incrementi, vel decrement! quadrat! velocitatum. Ad num. 189 inquire in appulsus curvse 
ad axem, sive is ibi secetur ab eadem (quo casu habentur transitus vel a repulsione ad 
attractionem, vel ab attractione ad repulsionem, quos dico limites, & quorum maximus est 
in tota mea Theoria usus), sive tangatur, & curva retro redeat, ubi etiam pro appulsibus 
considero recessus in infinitum per arcus asymptoticos, & qui transitus, sive limites, oriantur 
inde, vel in Natura admitti possint, evolvo. 

189 Num. 189 a consideratione curvae ad punctorum combinationem gradum facio, ac 

primo quidem usque ad num. 204 ago de systemate duorum punctorum, ea pertractans, 
quas pertinent ad eorum vires mutuas, & motus, sive sibi relinquantur, sive projiciantur 
utcunque, ubi & conjunctione ipsorum exposita in distantiis limitum, & oscillationibus 
variis, sive nullam externam punctorum aliorum actionem sentiant, sive perturbentur ab 
eadem, illud innuo in antecessum, quanto id usui futurum sit in parte tertia ad exponenda 
cohaesionis varia genera, fermentationes, conflagrationes, emissiones vaporum, proprietates 
luminis, elasticitatem, mollitiem. 

204 Succedit a Num. 204 ad 239 multo uberior consideratio trium punctorum, quorum 

vires generaliter facile definiuntur data ipsorum positione quacunque : verum utcunque 
data positione, & celeritate nondum a Geometris inventi sunt motus ita, ut generaliter pro 
casibus omnibus absolvi calculus possit. Vires igitur, & variationem ingentem, quam 
diversae pariunt combinationes punctorum, utut tantummodo numero trium, persequor 

209 usque ad num. 209. Hinc usque ad num. 214 quaedam evolvo, quae pertinent ad vires 
ortas in singulis ex actione composita reliquorum duorum, & quae tertium punctum non ad 
accessum urgeant, vel recessum tantummodo respectu eorundem, sed & in latus, ubi & 
soliditatis imago prodit, & ingens sane discrimen in distantiis particularum perquam exiguis 
ac summa in maximis, in quibus gravitas agit, conformitas, quod quanto itidem ad Naturae 

214 explicationem futurum sit usui, significo. Usque ad num. 221 ipsis etiam oculis contem- 
plandum propono ingens discrimen in legibus virium, quibus bina puncta agunt in tertium, 
sive id jaceat in recta, qua junguntur, sive in recta ipsi perpendiculari, & eorum intervallum 
secante bifariam, constructis ex data primigenia curva curvis vires compositas exhibentibus : 

221 turn sequentibus binis numeris casum evolvo notatu dignissimum, in quo mutata sola 
positione binorum punctorum, punctum tertium per idem quoddam intervallum, situm in 
eadem distantia a medio eorum intervallo, vel perpetuo attrahitur, vel perpetuo repellitur, 
vel nee attrahitur, nee repellitur ; cujusmodi discrimen cum in massis haberi debeat multo 

222 majus, illud indico, num. 222, quantus inde itidem in Physicam usus proveniat. 

223 Hie jam num. 223 a viribus binorum punctorum transeo ad considerandum totum 

ipsorum systema, & usque ad num. 228 contemplor tria puncta in directum sita, ex quorum 
mutuis viribus relationes quaedam exurgunt, quas multo generaliores redduntur inferius, ubi 
in tribus etiam punctis tantummodo adumbrantur, quae pertinent ad virgas rigidas, flexiles, 
elasticas, ac ad vectem, & ad alia plura, quae itidem inferius, ubi de massis, multo generaliora 

228 fiunt. Demum usque ad num. 238 contemplor tria puncta posita non in directum, sive in 
aequilibrio sint, sive in perimetro ellipsium quarundam, vel curvarum aliarum ; in quibus 
mira occurrit analogia limitum quorundam cum limitibus, quos habent bina puncta in axe 
curvae primigeniae ad se invicem, atque ibidem multo major varietas casuum indicatur pro 
massis, & specimen applicationis exhibetur ad soliditatem, & liquationem per celerem 

238 intestinum motum punctis impressum. Sequentibus autem binis numeris generalia quaedam 
expono de systemate punctorum quatuor cum applicatione ad virgas solidas, rigidas, flexiles, 
ac ordines particularum varies exhibeo per pyramides, quarum infimae ex punctis quatuor, 
superiores ex quatuor pyramidibus singulae coalescant. 

24 A num. 240 ad massas gradu facto usque a num. 264 considero, quae ad centrum gravi- 

tatis pertinent, ac demonstro generaliter, in quavis massa esse aliquod, & esse unicum : 
ostendo, quo pacto determinari generaliter possit, & quid in methodo, quae communiter 
adhibetur, desit ad habendam demonstrationis vim, luculenter expono, & suppleo, ac 


to Art. 179, I consider the areas included by the arcs; these, corresponding to different 172 
segments of the axis, may be of any magnitude whatever, either great or small ; moreover 
they measure the increment or decrement in the squares of the velocities. Then, on as 179 
far as Art. 189, 1 investigate the approach of the curve to the axis ; both when the former 
is cut by the latter, in which case there are transitions from repulsion to attraction and 
from attraction to repulsion, which I call ' limits,' & use very largely in every part of my 
Theory ; & also when the former is touched by the latter, & the curve once again recedes 
from the axis. I consider, too, as a case of approach, recession to infinity along an asymp- 
totic arc ; and I investigate what transitions, or limits, may arise from such a case, & 
whether such are admissible in Nature. 

In Art. 189, I pass on from the consideration of the curve to combinations of points. l %9 
First, as far as Art. 204, I deal with a system of two points. I work out those things that 
concern their mutual forces, and motions, whether they are left to themselves or pro- 
jected in any manner whatever. Here also, having explained the connection between 
these motions & the distances of the limits, & different cases of oscillations, whether they 
are affected by external action of other points, or are not so disturbed, I make an antici- 
patory note of the great use to which this will be put in the third part, for the purpose 
of explaining various kinds of cohesion, fermentations, conflagrations, emissions of vapours, 
the properties of light, elasticity and flexibility. 

There follows, from Art. 204 to Art. 239, the much more fruitful consideration of a 204 
system of three points. The forces connected with them can in general be easily deter- 
mined for any given positions of the points ; but, when any position & velocity are given, 
the motions have not yet been obtained by geometricians in such a form that the general 
calculation can be performed for every possible case. So I proceed to consider the forces, 
& the huge variation that different combinations of the points beget, although they are 
only three in number, as far as Art. 209. From that, on to Art. 214, I consider certain 209 
things that have to do with the forces that arise from the action, on each of the points, of 
the other two together, & how these urge the third point not only to approach, or recede 
from, themselves, but also in a direction at right angles ; in this connection there comes 
forth an analogy with solidity, & a truly immense difference between the several cases when 
the distances are very small, & the greatest conformity possible at very great distances 
such as those at which gravity acts ; & I point out what great use will be made of this also 
in explaining the constitution of Nature. Then up to Art. 221, I give ocular demonstra- 214 
tions of the huge differences that there are in the laws of forces with which two points act 
upon a third, whether it lies in the right line joining them, or in the right line that is the 
perpendicular which bisects the interval between them ; this I do by constructing, from 
the primary curve, curves representing the composite forces. Then in the two articles 221 
that follow, I consider the case, a really important one, in which, by merely changing the 
position of the two points, the third point, at any and the same definite interval situated 
at the same distance from the middle point of the interval between the two points, will 
be either continually attracted, or continually repelled, or neither attracted nor repelled ; 
& since a difference of this kind should hold to a much greater degree in masses, I point 
out, in Art. 222, the great use that will be made of this also in Physics. 222 

At this point then, in Art. 223, I pass from the forces derived from two points to the 22 3 
consideration of a whole system of them ; and, as far as Art. 228, I study three points 
situated in a right line, from the mutual forces of which there arise certain relations, which 
I return to later in much greater generality ; in this connection also are outlined, for three 
points only, matters that have to do with rods, either rigid, flexible or elastic, and with 
the lever, as well as many other things ; these, too, are treated much more generally later 
on, when I consider masses. Then right on to Art. 238, I consider three points that do 
not lie in a right line, whether they are in equilibrium, or moving in the perimeters of 
certain ellipses or other curves. Here we come across a marvellous analogy between certain 
limits and the limits which two points lying on the axis of the primary curve have with 
respect to each other ; & here also a much greater variety of cases for masses is shown, 
& an example is given of the application to solidity, & liquefaction, on account of a quick 
internal motion being impressed on the points of the body. Moreover, in the two articles 
that then follow, I state some general propositions with regard to a system of four points, 
together with their application to solid rods, both rigid and flexible ; I also give an illus- 
tration of various classes of particles by means of pyramids, each of which is formed of four 
points in the most simple case, & of four of such pyramids in the more complicated cases. 

From Art. 240 as far as Art. 264, I pass on to masses & consider matters pertaining to 2 4 
the centre of gravity ; & I prove that in general there is one, & only one, in any given mass. 
I show how it can in general be determined, & I set forth in clear terms the point that is 
lacking in the usual method, when it comes to a question of rigorous proof ; this deficiency 


exemplum profero quoddam ejusdem generis, quod ad numerorum pertinet multiplica- 
tionem, & ad virium compositionem per parallelogramma, quam alia methodo generaliore 
exhibeo analoga illi ipsi, qua generaliter in centrum gravitatis inquire : turn vero ejusdem 
ope demonstro admodum expedite, & accuratissime celebre illud Newtoni theorema de 
statu centri gravitatis per mutuas internas vires numquam turbato. 

264 Ejus tractionis fructus colligo plures : conservationem ejusdem quantitatis motuum in 

265 Mundo in eandem plagam num. 264, sequalitatem actionis, & reactionis in massis num. 265, 

266 collisionem corporum, & communicationem motus in congressibus directis cum eorum 
276 legibus, inde num. 276 congressus obliques, quorum Theoriam a resolutione motuum reduce 

277, 278 ad compositionem num. 277, quod sequent! numero 278 transfero ad incursum etiam in 
270 planum immobile ; ac a num. 279 ad 289 ostendo nullam haberi in Natura veram virium, 
aut motuum resolutionem, sed imaginariam tantummodo, ubi omnia evolvo, & explico 
casuum genera, quae prima fronte virium resolutionem requirere videntur. 

289 A num. 289 ad 297 leges expono compositionis virium, & resolutionis, ubi & illud 

notissimum, quo pacto in compositione decrescat vis, in resolutione crescat, sed in ilia priore 
conspirantium summa semper maneat, contrariis elisis ; in hac posteriore concipiantur 
tantummodo binae vires contrarise adjectas, quse consideratio nihil turbet phenomena ; 
unde fiat, ut nihil inde pro virium vivarum Theoria deduci possit, cum sine iis explicentur 
omnia, ubi plura itidem explico ex iis phsenomenis, quse pro ipsis viribus vivis afferri solent. 

2Q7 A num. 297 occasione inde arrepta aggredior qusedam, quae ad legem continuitatis 

pertinent, ubique in motibus sancte servatam, ac ostendo illud, idcirco in collisionibus 
corporum, ac in motu reflexo, leges vulgo definitas, non nisi proxime tantummodo observari, 
& usque ad num. 307 relationes varias persequor angulorum incidentisa, & reflexionis, sive 
vires constanter in accessu attrahant, vel repellant constanter, sive jam attrahant, jam 
repellant : ubi & illud considero, quid accidat, si scabrities superficiei agentis exigua sit, 
quid, si ingens, ac elementa profero, quae ad luminis reflexionem, & refractionem explican- 
dam, definiendamque ex Mechanica requiritur, relationem itidem vis absolutae ad relativam 
in obliquo gravium descensu, & nonnulla, quae ad oscillationum accuratiorem Theoriam 
necessaria sunt, prorsus elementaria, diligenter expono. 

307 A num. 307 inquire in trium massarum systema, ubi usque ad num. 313 theoremata 

evolvo plura, quae pertinent ad directionem virium in singulis compositarum e binis 
reliquarum actionibus, ut illud, eas directiones vel esse inter se parallelas, vel, si utrinque 

313 indefinite producantur, per quoddam commune punctum transire omnes : turn usque ad 
321 theoremata alia plura, quae pertinent ad earumdem compositarum virium rationem ad 
se invicem, ut illud & simplex, & elegans, binarum massarum vires acceleratrices esse semper 
in ratione composita ex tribus reciprocis rationibus, distantise ipsarum a massa tertia, sinus 
anguli, quern singularum directio continet cum sua ejusmodi distantia, & massae ipsius earn 
habentis compositam vim, ad distantiam, sinum, massam alteram ; vires autem motrices 
habere tantummodo priores rationes duas elisa tertia. 

321 Eorum theorematum fructum colligo deducens inde usque ad num. 328, quae ad 

aequilibrium pertinent divergentium utcumque virium, & ipsius aequilibrii centrum, ac 
nisum centri in fulcrum, & quae ad prseponderantiam, Theoriam extendens ad casum etiam, 
quo massae non in se invicem agant mutuo immediate, sed per intermedias alias, quse nexum 
concilient, & virgarum nectentium suppleant vices, ac ad massas etiam quotcunque, quarum 
singulas cum centro conversionis, & alia quavis assumpta massa connexas concipio, unde 
principium momenti deduce pro machinis omnibus : turn omnium vectium genera evolvo, 
ut & illud, facta suspensione per centrum gravitatis haberi aequilibrium, sed in ipso centro 
debere sentiri vim a fulcro, vel sustinente puncto, sequalem summae ponderum totius 
systematis, unde demum pateat ejus ratio, quod passim sine demonstratione assumitur, 
nimirum systemate quiescente, & impedito omni partium motu per aequilibrium, totam 
massam concipi posse ut in centro gravitatis collectam. 


I supply, & I bring forward a certain example of the same sort, that deals with the multi- 
plication of numbers, & to the composition of forces by the parallelogram law ; the latter 
I prove by another more general method, analogous to that which I use in the general 
investigation for the centre of gravity. Then by its help I prove very expeditiously & 
with extreme rigour that well-known theorem of Newton, in which he affirmed that the 
state of the centre of gravity is in no way altered by the internal mutual forces. 

I gather several good results from this method of treatment. In Art. 264, the con- 264 
servation of the same quantity of motion in the Universe in one plane ; in Art. 265 the 265 
equality of action and reaction amongst masses ; then the collision of solid bodies, and the 266 
communication of motions in direct impacts & the laws that govern them, & from that, 276 
in Art. 276, oblique impacts ; in Art. 277 I reduce the theory of these from resolution of 277 
motions to compositions, & in the article that follows, Art. 278, I pass to impact on to a 278 
fixed plane; from Art. 279 to Art. 289 I show that there can be no real resolution of forces 279 
or of motions in Nature, but only a hypothetical one ; & in this connection I consider & 
explain all sorts of cases, in which at first sight it would seem that there must be resolution. 

From Art. 289 to Art. 297, 1 state the laws for the composition & resolution of forces ; 289 
here also I give the explanation of that well-known fact, that force decreases in composition, 
increases in resolution, but always remains equal to the sum of the parts acting in the same 
direction as itself in the first, the rest being equal & opposite cancel one another ; whilst 
in the second, all that is done is to suppose that two equal & opposite forces are added on, 
which supposition has no effect on the phenomena. Thus it comes about that nothing 
can be deduced from this in favour of the Theory of living forces, since everything can be 
explained without them ; in the same connection, I explain also many of the phenomena, 
which are usually brought forward as evidence in favour of these ' living forces.' 

In Art. 297, I seize the opportunity offered by the results just mentioned to attack 207 
certain matters that relate to the law of continuity, which in all cases of motion is strictly 
observed ; & I show that, in the collision of solid bodies, & in reflected motion, the laws, 
as usually stated, are therefore only approximately followed. From this, as far as Art. 307, 
I make out the various relations between the angles of incidence & reflection, whether the 
forces, as the bodies approach one another, continually attract, or continually repel, or 
attract at one time & repel at another. I also consider what will happen if the roughness 
of the acting surface is very slight, & what if it is very great. I also state the first principles, 
derived from mechanics, that are required for the explanation & determination of the 
reflection & refraction of light ; also the relation of the absolute to the relative force in 
the oblique descent of heavy bodies ; & some theorems that are requisite for the more 
accurate theory of oscillations ; these, though quite elementary, I explain with great care. 

From Art. 307 onwards, I investigate the system of three bodies ; in this connection, 
as far as Art. 313, I evolve several theorems dealing with the direction of the forces on each 
one of the three compounded from the combined actions of the other two ; such as the 
theorem, that these directions are either all parallel to one another, or all pass through 
some one common point, when they are produced indefinitely on both sides. Then, as ^j, 
far as Art. 321, I make out several other theorems dealing with the ratios of these same 
resultant forces to one another ; such as the following very simple & elegant theorem, that 
the accelerating forces of two of the masses will always be in a ratio compounded of three 
reciprocal ratios ; namely, that of the distance of either one of them from the third mass, 
that of the sine of the angle which the direction of each force makes with the corresponding 
distance of this kind, & that of the mass itself on which the force is acting, to the corre- 
sponding distance, sine and mass for the other : also that the motive forces only have the 
first two ratios, that of the masses being omitted. 

I then collect the results to be derived from these theorems, deriving from them, as far , 2I 
as Art. 328, theorems relating to the equilibrium of forces diverging in any manner, & the 
centre of equilibrium, & the pressure of the centre on a fulcrum. I extend the theorem 
relating to preponderance to the case also, in which the masses do not mutually act upon 
one another in a direct manner, but through others intermediate between them, which 
connect them together, & supply the place of rods joining them ; and also to any number of 
masses, each of which I suppose to be connected with the centre of rotation & some other 
assumed mass, & from this I derive the principles of moments for all machines. Then I 
consider all the different kinds of levers ; one of the theorems that I obtain is, that, if a 
lever is suspended from the centre of gravity, then there is equilibrium ; but a force should 
be felt in this centre from the fulcrum or sustaining point, equal to the sum of the weights 
of the whole system ; from which there follows most clearly the reason, which is every- 
where assumed without proof, why the whole mass can be supposed to be collected at its 
centre of gravity, so long as the system is in a state of rest & all motions of its parts are pro- 
hibited by equilibrium. 


328 A num. 328 ad 347 deduce ex iisdem theorematis, quae pertinent ad centrum oscilla- 

tionis quotcunque massarum, sive sint in eadem recta, sive in piano perpendiculari ad axem 
rotationis ubicunque, quse Theoria per systema quatuor massarum, excolendum aliquanto 
diligentius, uberius promoveri deberet & extendi ad generalem habendum solidorum nexum, 

344 qua re indicata, centrum itidem percussionis inde evolve, & ejus analogiam cum centre 
oscillationis exhibeo. 

347 Collecto ejusmodi fructu ex theorematis pertinentibus ad massas tres, innuo num. 347, 
quae mihi communia sint cum ceteris omnibus, & cum Newtonianis potissimum, pertinentia 
ad summas virium, quas habet punctum, vel massa attracta, vel repulsa a punctis singulis 

348 alterius massae ; turn a num. 348 ad finem hujus partis, sive ad num. 358, expono quasdam, 
quae pertinent ad fluidorum Theoriam, & primo quidem ad pressionem, ubi illud innuo 
demonstratum a Newtono, si compressio fluidi sit proportionalis vi comprimenti, vires 
repulsivas punctorum esse in ratione reciproca distantiarum, ac vice versa : ostendo autem 
illud, si eadem vis sit insensibilis, rem, praeter alias curvas, exponi posse per Logisticam, 
& in fluidis gravitate nostra terrestri prseditis pressiones haberi debere ut altitudines ; 
deinde vero attingo ilia etiam, quae pertinent ad velocitatem fluidi erumpentis e vase, & 
expono, quid requiratur, ut ea sit sequalis velocitati, quae acquiretur cadendo per altitudinem 
ipsam, quemadmodum videtur res obtingere in aquae efHuxu : quibus partim expositis, 
partim indicatis, hanc secundam partem conclude. 


358 Num. 358 propono argumentum hujus tertise partis, in qua omnes e Theoria mea 

360 generales materis proprietates deduce, & particulares plerasque : turn usque ad num. 371 
ago aliquanto fusius de impenetrabilitate, quam duplicis generis agnosco in meis punctorum 
inextensorum massis, ubi etiam de ea apparenti quadam compenetratione ago, ac de luminis 
trarlsitu per substantias intimas sine vera compenetratione, & mira quaedam phenomena 

371 hue pertinentia explico admodum expedite. Inde ad num. 375 de extensione ago, quae 
mihi quidem in materia, & corporibus non est continua, sed adhuc eadem praebet phaeno- 
menae sensibus, ac in communi sententia ; ubi etiam de Geometria ago, quae vim suam in 

375 mea Theoria retinet omnem : turn ad num. 383 figurabilitatem perseqUor, ac molem, 
massam, densitatem singillatim, in quibus omnibus sunt quaedam Theoriae meae propria 

383 scitu non indigna. De Mobilitate, & Motuum Continuitate, usque ad num. 388 notatu 

388 digna continentur : turn usque ad num. 391 ago de aequalitate actionis, & reactionis, cujus 
consectaria vires ipsas, quibus Theoria mea innititur, mirum in modum conformant. 
Succedit usque ad num. 398 divisibilitas, quam ego ita admitto, ut quaevis massa existens 
numerum punctorum realium habeat finitum tantummodo, sed qui in data quavis mole 
possit esse utcunque magnus ; quamobrem divisibilitati in infinitum vulgo admissae sub- 
stituo componibilitatem in infinitum, ipsi, quod ad Naturae phenomena explicanda 

398 pertinet, prorsus aequivalentem. His evolutis addo num. 398 immutabilitatem primorum 
materiae elementorum, quse cum mihi sint simplicia prorsus, & inextensa, sunt utique 
immutabilia, & ad exhibendam perennem phasnomenorum seriem aptissima. 

399 A num. 399 ad 406 gravitatem deduco ex mea virium Theoria, tanquam ramum 

quendam e communi trunco, ubi & illud expono, qui fieri possit, ut fixae in unicam massam 

406 non coalescant, quod gravitas generalis requirere videretur. Inde ad num. 419 ago de 
cohaesione, qui est itidem veluti alter quidam ramus, quam ostendo, nee in quiete con- 
sistere, nee in motu conspirante, nee in pressione fluidi cujuspiam, nee in attractione 
maxima in contactu, sed in limitibus inter repulsionem, & attractionem ; ubi & problema 
generale propono quoddam hue pertinens, & illud explico, cur massa fracta non iterum 
coalescat, cur fibrae ante fractionem distendantur, vel contrahantur, & innuo, quae ad 
cohaesionem pertinentia mihi cum reliquis Philosophis communia sint. 

419 A cohacsione gradum facio num. 419 ad particulas, quae ex punctis cohaerentibus 

efformantur, de quibus .ago usque ad num. 426. & varia persequor earum discrimina : 


From Art. 328 to Art. 347, I deduce from these same theorems, others that relate to 328 
the centre of oscillation of any number of masses, whether they are in the same right line, 
or anywhere in a plane perpendicular to the axis of rotation ; this theory wants to be worked 
somewhat more carefully with a system of four bodies, to be gone into more fully, & to 
be extended so as to include the general case of a system of solid bodies ; having stated 
this, I evolve from it the centre of percussion, & I show the analogy between it & the centre 344 
of oscillation. 

I obtain all such results from theorems relating to three masses. After that, in Art. 347 
347, I intimate the matters in which I agree with all others, & especially with the followers 
of Newton, concerning sums of forces, acting on a point, or an attracted or repelled mass, 
due to the separate points of another mass. Then, from Art. 348 to the end of this part, 348 
i.e., as far as Art. 359, I expound certain theorems that belong to the theory of fluids ; & 
first of all, theorems with regard to pressure, in connection with which I mention that one 
which was proved by Newton, namely, that, if the compression of a fluid is proportional to 
the compressing force, then the repulsive forces between the points are in the reciprocal 
ratio of the distances, & conversely. Moreover, I show that, if the same force is insen- 
sible, then the matter can be represented by the logistic & other curves ; also that in fluids 
subject to our terrestrial gravity pressures should be found proportional to the depths. 
After that, I touch upon those things that relate to the velocity of a fluid issuing from a 
vessel ; & I show what is necessary in order that this should be equal to the velocity which 
would be acquired by falling through the depth itself, just as it is seen to happen in the 
case of an efflux of water. These things in some part being explained, & in some part 
merely indicated, I bring this second part to an end. 


In Art. 358, I state the theme of this third part ; in it I derive all the general & most 358 
of the special, properties of matter from my Theory. Then, as far as Art. 371, I deal some- 360 
what more at length with the subject of impenetrability, which I remark is of a twofold 
kind in my masses of non-extended points ; in this connection also, I deal with a certain 
apparent case of compenetrability, & the passage of light through the innermost parts of 
bodies without real compenetration ; I also explain in a very summary manner several 
striking phenomena relating to the above. From here on to Art. 375, I deal with exten- 371 
sion ; this in my opinion is not continuous either in matter or in solid bodies, & yet it 
yields the same phenomena to the senses as does the usually accepted idea of it ; here I 
also deal with geometry, which conserves all its power under my Theory. Then, as far 375 
as Art. 383, I discuss figurability, volume, mass & density, each in turn ; in all of these 
subjects there are certain special points of my Theory that are not unworthy of investi- 
gation. Important theorems on mobility & continuity of motions are to be found from 
here on to Art. 388 ; then, as far as Art. 391, I deal with the equality of action & reaction, 
& my conclusions with regard to the subject corroborate in a wonderful way the hypothesis 
of those forces, upon which my Theory depends. Then follows divisibility, as far as Art. 39 1 
398 ; this principle I admit only to the extent that any existing mass may be made up of 
a number of real points that are finite only, although in any given mass this finite number 
may be as great as you please. Hence for infinite divisibility, as commonly accepted, I 
substitute infinite multiplicity ; which comes to exactly the same thing, as far as it is 
concerned with the explanation of the phenomena of Nature. Having considered these 
subjects I add, in Art. 398, that of the immutability of the primary elements of matter ; 398 
according to my idea, these are quite simple in composition, of no extent, they are every- 
where unchangeable, & hence are splendidly adapted for explaining a continually recurring 
set of phenomena. 

From Art. 399 to Art. 406, 1 derive gravity from my Theory of forces, as if it were a 399 
particular branch on a common trunk ; in this connection also I explain how it can happen 
that the fixed stars do not all coalesce into one mass, as would seem to be required under 406 
universal gravitation. Then, as far as Art. 419, I deal with cohesion, which is also as it 
were another branch ; I show that this is not dependent upon quiescence, nor on motion 
that is the same for all parts, nor on the pressure of some fluid, nor on the idea that the 
attraction is greatest at actual contact, but on the limits between repulsion and attraction. 
I propose, & solve, a general problem relating to this, namely, why masses, once broken, 
do not again stick together, why the fibres are stretched or contracted before fracture 
takes place ; & I intimate which of my ideas relative to cohesion are the same as those 
held by other philosophers. 

In Art. 419, 1 pass on from cohesion to particles which are formed from a number of 4 J 9 
cohering points ; & I consider these as far as Art. 426, & investigate the various distinctions 


ostendo nimirum, quo pacto varias induere possint figuras quascunque, quarum tenacissime 
sint ; possint autem data quavis figura discrepare plurimum in numero, & distributione 
punctorum, unde & oriantur admodum inter se diversae vires unius particulae in aliam, ac 
itidem diversae in diversis partibus ejusdem particulae respectu diversarum partium, vel 
etiam respectu ejusdem partis particulse alterius, cum a solo numero, & distributione 
punctorum pendeat illud, ut data particula datam aliam in datis earum distantiis, & 
superficierum locis, vel attrahat, vel repellat, vel respectu ipsius sit prorsus iners : turn illud 
addo, particulas eo dimcilius dissolubiles esse, quo minores sint ; debere autem in gravitate 
esse penitus uniformes, quaecunque punctorum dispositio habeatur, & in aliis proprietatibus 
plerisque debere esse admodum (uti observamus) diversas, quae diversitas multo major in 
majoribus massis esse debeat. 

426 A num. 426 ad 446 de solidis, & fluidis, quod discrimen itidem pertinet ad varia 

cohaesionum genera ; & discrimen inter solida, & fluida diligenter expono, horum naturam 
potissimum repetens ex motu faciliori particularum in gyrum circa alias, atque id ipsum ex 
viribus circumquaque aequalibus ; illorum vero ex inaequalitate virium, & viribus quibusdam 
in latus, quibus certam positionem ad se invicem servare debeant. Varia autem distinguo 
fluidorum genera, & discrimen profero inter virgas rigidas, flexiles, elasticas, fragiles, ut & 
de viscositate, & humiditate ago, ac de organicis, & ad certas figuras determinatis corporibus, 
quorum efformatio nullam habet difficultatem, ubi una particula unam aliam possit in 
certis tantummodo superficiei partibus attrahere, & proinde cogere ad certam quandam 
positionem acquirendam respectu ipsius, & retinendam. Demonstro autem & illud, posse 
admodum facile ex certis particularum figuris, quarum ipsae tenacissimae sint, totum etiam 
Atomistarum, & Corpuscularium systema a mea Theoria repeti ita, ut id nihil sit aliud, 
nisi unicus itidem hujus veluti trunci foecundissimi ramus e diversa cohaesionis ratione 
prorumpens. Demum ostendo, cur non quaevis massa, utut constans ex homogeneis 
punctis, & circa se maxime in gyrum mobilibus, fluida sit ; & fluidorum resistentiam quoque 
attingo, in ejus leges inquirens. 

446 A num. 446 ad 450 ago de iis, quae itidem ad diversa pertinent soliditatis genera, nimirum 

de elasticis, & mollibus, ilia repetens a magna inter limites proximos distantia, qua fiat, ut 
puncta longe dimota a locis suis, idem ubique genus virium sentiant, & proinde se ad 
priorem restituant locum ; hasc a limitum frequentia, atque ingenti vicinia, qua fiat, ut ex 
uno ad alium delata limitem puncta, ibi quiescant itidem respective, ut prius. Turn vero 
de ductilibus, & malleabilibus ago, ostendens, in quo a fragilibus discrepent : ostendo autem, 
haec omnia discrimina a densitate nullo modo pendere, ut nimirum corpus, quod 
multo sit altero densius, possit tarn multo majorem, quam multo minorem soliditatem, & 
cohaesionem habere, & quaevis ex proprietatibus expositis aeque possit cum quavis vel majore, 
vel minore densitate componi. 

450 Num. 450 inquire in vulgaria quatuor elementa ; turn a num. 451 ad num. 467 persequor 

452 chemicas operationes ; num. 452 explicans dissolutionem, 453 praecipitationem, 454, & 455 
commixtionem plurium substantiarum in unam : turn num. 456, & 457 liquationem binis 
methodis, 458 volatilizationem, & effervescentiam, 461 emissionem efHuviorum, quae e massa 
constanti debeat esse ad sensum constans, 462 ebullitionem cum variis evaporationum 
generibus ; 463 deflagrationem, & generationem aeris ; 464 crystallizationem cum certis 
figuris ; ac demum ostendo illud num. 465, quo pacto possit fermentatio desinere ; & num. 
466, quo pacto non omnia fermentescant cum omnibus. 

467 A fermentatione num. 467 gradum facio ad ignem, qui mihi est fermentatio quaedam 

substantiae lucis cum sulphurea quadam substantia, ac plura inde consectaria deduce usque 

471 ad num. 471 ; turn ab igne ad lumen ibidem transeo, cujus proprietates praecipuas, ex 

472 quibus omnia lucis phaenomena oriuntur, propono num. 472, ac singulas a Theoria mea 
deduce, & fuse explico usque ad num. 503, nimirum emissionem num. 473, celeritatem 474, 
propagationem rectilineam per media homogenea, & apparentem tantummodo compene- 
trationem a num. 475 ad 483, pellucidatem, & opacitatem num. 483, reflexionem ad angulos 
aequales inde ad 484, refractionem ad 487, tenuitatem num. 487, calorem, & ingentes 
intestines motus allapsu tenuissimae lucis genitos, num. 488, actionem majorem corporum 
eleosorum, & sulphurosorum in lumen num. 489 : turn num. 490 ostendo, nullam resist- 


between them. I show how it is possible for various shapes of all sorts to be assumed, 
which offer great resistance to rupture ; & how in a given shape they may differ very greatly 
in the number & disposition of the points forming them. Also that from this fact there 
arise very different forces for the action of one particle upon another, & also for the action 
of different parts of this particle upon other different parts of it, or on the same part of 
another particle. For that depends solely on the number & distribution of the points, 
so that one given particle either attracts, or repels, or is perfectly inert with regard to 
another given particle, the distances between them and the positions of their surfaces being 
also given. Then I state in addition that the smaller the particles, the greater is the diffi- 
culty in dissociating them ; moreover, that they ought to be quite uniform as regards 
gravitation, no matter what the disposition of the points may be ; but in most other 
properties they should be quite different from one another (which we observe to be the 
case) ; & that this difference ought to be much greater in larger masses. 

From Art. 426 to Art. 446, 1 consider solids & fluids, the difference between which is 426 
also a matter of different kinds of cohesion. I explain with great care the difference 
between solids & fluids ; deriving the nature of the latter from the greater freedom of motion 
of the particles in the matter of rotation about one another, this being due to the forces 
being nearly equal ; & that of the former from the inequality of the forces, and from certain 
lateral forces which help them to keep a definite position with regard to one another. I 
distinguish between various kinds of fluids also, & I cite the distinction between rigid, 
flexible, elastic & fragile rods, when I deal with viscosity & humidity ; & also in dealing with 
organic bodies & those solids bounded by certain fixed figures, of which the formation 
presents no difficulty ; in these one particle can only attract another particle in certain 
parts of the surface, & thus urge it to take up some definite position with regard to itself, 
& keep it there. I also show that the whole system of the Atomists, & also of the Corpus- 
cularians, can be quite easily derived by my Theory, from the idea of particles of definite 
shape, offering a high resistance to deformation ; so that it comes to nothing else than 
another single branch of this so to speak most fertile trunk, breaking forth from it 
on account of a different manner of cohesion. Lastly, I show the reason why it is that 
not every mass, in spite of its being constantly made up of homogeneous points, & even 
these in a high degree capable of rotary motion about one another, is a fluid. I also touch 
upon the resistance of fluids, & investigate the laws that govern it. 

From Art. 446 to Art. 450, I deal with those things that relate to the different kinds 446 
of solidity, that is to say, with elastic bodies, & those that are soft. I attribute the nature 
of the former to the existence of a large interval between the consecutive limits, on account 
of which it comes about that points that are far removed from their natural positions still 
feel the effects of the same kind of forces, & therefore return to their natural positions ; 
& that of the latter to the frequency & great closeness of the limits, on account of which it 
comes about that points that have been moved from one limit to another, remain there 
in relative rest as they were to start with. Then I deal with ductile and malleable solids, 
pointing out how they differ from fragile solids. Moreover I show that all these differ- 
ences are in no way dependent on density ; so that, for instance, a body that is much more 
dense than another body may have either a much greater or a much less solidity and 
cohesion than another ; in fact, any of the properties set forth may just as well be combined 
with any density either greater or less. 

In Art. 450 I consider what are commonly called the " four elements " ; then from 450 
Art. 451 to Art. 467, I treat of chemical operations ; I explain solution in Art. 452, preci- 452 
pitation in Art. 453, the mixture of several substances to form a single mass in Art. 454, 
455, liquefaction by two methods in Art. 456, 457, volatilization & effervescence in Art. 
458, emission of effluvia (which from a constant mass ought to be approximately constant) 
in. Art. 461, ebullition & various kinds of evaporation in Art. 462, deflagration & generation 
of gas in Art. 463, crystallization with definite forms of crystals in Art. 464 ; & lastly, I show, 
in Art. 465, how it is possible for fermentation to cease, & in Art. 466, how it is that any 
one thing does not ferment when mixed with any other thing. 

From fermentation I pass on, in Art. 467, to fire, which I look upon as a fermentation 467 
of some substance in light with some sulphureal substance ; & from this I deduce several 
propositions, up to Art. 471. There I pass on from fire to light, the chief properties of 471 
which, from which all the phenomena of light arise, I set forth in Art. 472 ; & I deduce 472 
& fully explain each of them in turn as far as Art. 503. Thus, emission in Art. 473, velo- 
city in Art. 474, rectilinear propagation in homogeneous media, & a compenetration that 
is merely apparent, from Art. 475 on to Art. 483, pellucidity & opacity in Art. 483, reflec- 
tion at equal angles to Art. 484, & refraction to Art. 487, tenuity in Art. 487, heat & the 
great internal motions arising from the smooth passage of the extremely tenuous light in 
Art. 488, the greater action of oleose & sulphurous bodies on light in Art. 489. Then I 


entiam veram pati, ac num. 491 explico, unde sint phosphora, num. 492 cur lumen cum 
majo e obliquitate incidens reflectatur magis, num. 493 & 494 unde diversa refrangibilitas 
ortum ducat, ac num. 495, & 496 deduce duas diversas dispositiones ad asqualia redeuntes 
intervalla, unde num. 497 vices illas a Newtono detectas facilioris reflexionis, & facilioris 
transmissus eruo, & num. 498 illud, radios alios debere reflecti, alios transmitti in appulsu 
ad novum medium, & eo plures reflecti, quo obliquitas incidentise sit major, ac num. 
499 & 500 expono, unde discrimen in intervallis vicium, ex quo uno omnis naturalium 
colorum pendet Newtoniana Theoria. Demum num. 501 miram attingo crystalli 
Islandicse proprietatem, & ejusdem causam, ac num. 502 diffractionem expono, quse est 
quaedam inchoata refractio, sive reflexio. 

503 Post lucem ex igne derivatam, quse ad oculos pertinet, ago brevissime num. 503 de 

504 sapore, & odore, ac sequentibus tribus numeris de sono : turn aliis quator de tactu, ubi 
507 etiam de frigore, & calore : deinde vero usque ad num. 514 de electricitate, ubi totam 
511 Franklinianam Theoriam ex meis principiis explico, eandem ad bina tantummodo reducens 

principia, quse ex mea generali virium Theoria eodem fere pacto deducuntur, quo prsecipi- 
514 tationes, atque dissolutiones. Demum num. 514, ac 515 magnetismum persequor, tam 
directionem explicans, quam attractionem magneticam. 

516 Hisce expositis, quas ad particulares .etiam proprietates pertinent, iterum a num. 516 

ad finem usque generalem corporum complector naturam, & quid materia sit, quid forma, 
quse censeri debeant essentialia, quse accidentialia attributa, adeoque quid transformatio 
sit, quid alteratio, singillatim persequor, & partem hanc tertiam Theorise mesa absolve. 

De Appendice ad Metaphysicam pertinente innuam hie illud tantummodo, me ibi 
exponere de anima illud inprimis, quantum spiritus a materia differat, quern nexum anima 
habeat cum corpore, & quomodo in ipsum agat : turn de DEO, ipsius & existentiam me 
pluribus evincere, quae nexum habeant cum ipsa Theoria mea, & Sapientiam inprimis, ac 
Providentiam, ex qua gradum ad revelationem faciendum innuo tantummodo. Sed hsec 
in antecessum veluti delibasse sit satis. 


show, in Art. 490, that it suffers no real resistance, & in Art. 491 I explain the origin of 
bodies emitting light, in Art. 492 the reason why light that falls with greater obliquity 
is reflected more strongly, in Art. 493, 494 the origin of different degrees of refrangibility, 
& in Art. 495, 496 I deduce that there are two different dispositions recurring at equal 
intervals ; hence, in Art. 497, I bring out those alternations, discovered by Newton, of 
easier reflection & easier transmission, & in Art. 498 I deduce that some rays should be 
reflected & others transmitted in the passage to a fresh medium, & that the greater the obli- 
quity of incidence, the greater the number of reflected rays. In Art. 499, 500 I state the 
origin of the difference between the lengths of the intervals of the alternations ; upon this 
alone depends the whole of the Newtonian theory of natural colours. Finally, in Art. 501, 
I touch upon the wonderful property of Iceland spar & its cause, & in Art. 502 I explain 
diffraction, which is a kind of imperfect refraction or reflection. 

After light derived from fire, which has to do with vision, I very briefly deal with 
taste & smell in Art. 503, of sound in the three articles that follow next. Then, in the S3 
next four articles, I consider touch, & in connection with it, cold & heat also. After that, 54 
as far as Art. 514, I deal with electricity ; here I explain the whole of the Franklin theory 57 
by means of my principles ; I reduce this theory to two principles only, & these are 5 1 1 
derived from my general Theory of forces in almost the same manner as I have already derived 
precipitations & solutions. Finally, in Art. 514, 515, I investigate magnetism, explaining 5H 
both magnetic direction attraction. 

These things being expounded, all of which relate to special properties, I once more 
consider, in the articles from 516 to the end, the general nature of bodies, what matter is, 516 
its form, what things ought to be considered as essential, & what as accidental, attributes ; 
and also the nature of transformation and alteration are investigated, each in turn ; & 
thus I bring to a close the third part of my Theory. 

I will mention here but this one thing with regard to the appendix on Metaphysics ; 
namely, that I there expound more especially how greatly different is the soul from matter, 
the connection between the soul & the body, & the manner of its action upon it. Then 
with regard to GOD, I prove that He must exist by many arguments that have a close con- 
nection with this Theory of mine ; I especially mention, though but slightly, His Wisdom 
and Providence, from which there is but a step to be made towards revelation. But I think 
that I have, so to speak, given my preliminary foretaste quite sufficiently. 


In quo conveniat 
cum systemate 
Newtoniano, & 

Cujusmodi systema> 
Theoria exhibeat. 


Theorice expositio, analytica deductio^ & vindicatio. 

lRIUM mutuarum Theoria, in quam incidi jam ab Anno 1745, dum e 
notissimis principiis alia ex aliis consectaria eruerem, & ex qua ipsam 
simplicium materise elementorum constitutionem deduxi, systema 
exhibet medium inter Leibnitianum, & Newtonianum, quod nimirum 
& ex utroque habet plurimum, & ab utroque plurimum dissidet ; at 
utroque in immensum simplicius, proprietatibus corporum generalibus 
sane omnibus, & [2] peculiaribus quibusque praecipuis per accuratissimas 
demonstrationes deducendis est profecto mirum in modum idoneum. 

2. Habet id quidem ex Leibnitii Theoria elementa prima simplicia, ac prorsus inex- 
tensa : habet ex Newtoniano systemate vires mutuas, quae pro aliis punctorum distantiis a 
se invicem aliae sint ; & quidem ex ipso itidem Newtono non ejusmodi vires tantummodo, 
quse ipsa puncta determinent ad accessum, quas vulgo attractiones nominant ; sed etiam 
ejusmodi, quae determinent ad recessum, & appellantur repulsiones : atque id ipsum ita, 
ut, ubi attractio desinat, ibi, mutata distantia, incipiat repulsio, & vice versa, quod nimirum 
Newtonus idem in postrema Opticse Quaestione proposuit, ac exemplo transitus a positivis 
ad negativa, qui habetur in algebraicis formulis, illustravit. Illud autem utrique systemati 
commune est cum hoc meo, quod quaevis particula materiae cum aliis quibusvis, utcunque 
remotis, ita connectitur, ut ad mutationem utcunque exiguam in positione unius cujusvis, 
determinationes ad motum in omnibus reliquis immutentur, & nisi forte elidantur omnes 
oppositas, qui casus est infinities improbabilis, motus in iis omnibus aliquis inde ortus 

In quo differat a 
Leibnitiano & ipsi 

3. Distat autem a Leibnitiana Theoria longissime, turn quia nullam extensionem 
continuam admittit, quae ex contiguis, & se contingentibus inextensis oriatur : in quo 
quidem dirficultas jam olim contra Zenonem proposita, & nunquam sane aut soluta satis, 
aut solvenda, de compenetratione omnimoda inextensorum contiguorum, eandem vim 
adhuc habet contra Leibnitianum systema : turn quia homogeneitatem admittit in elementis, 
omni massarum discrimine a sola dispositione, & diversa combinatione derivato, ad quam 
homogeneitatem in elementis, & discriminis rationem in massis, ipsa nos Naturae analogia 
ducit, ac chemicae resolutiones inprimis, in quibus cum ad adeo pauciora numero, & adeo 
minus inter se diversa principiorum genera, in compositorum corporum analysi deveniatur, 
id ipsum indicio est, quo ulterius promoveri possit analysis, eo ad majorem simplicitatem, 
& homogeneitatem devenire debere, adeoque in ultima demum resolutione ad homogenei- 
tatem, & simplicitatem summam, contra quam quidem indiscernibilium principium, & 
principium rationis sufficients usque adeo a Leibnitianis depraedicata, meo quidem judicio, 
nihil omnino possunt. 

in quo differat a A Distat itidem a Newtoniano systemate quamplunmum, turn in eo, quod ea, quae 

Newtoniano & ipsi XT . . r\ r\ r 

praestet. Newtonus in ipsa postremo (Juaestione (Jpticae conatus est expncare per tna pnncipia, 

gravitatis, cohsesionis, fermentationis, immo & reliqua quamplurima, quae ab iis tribus 
principiis omnino non pendent, per unicam explicat legem virium, expressam unica, & ex 
pluribus inter se commixtis non composita algebraica formula, vel unica continua geometrica 
curva : turn in eo, quod in mi-[3]-nimis distantiis vires admittat non positivas, sive 
attractivas, uti Newtonus, sed negativas, sive repulsivas, quamvis itidem eo majores in 




Exposition ^ ^Analytical Derivation & Proof of the Theory 

I. ' ^i ^^ HE following Theory of mutual forces, which I lit upon as far back as the year Th e kind of sys- 
1745, whilst I was studying various propositions arising from other very p^ents. 6 
well-known principles, & from which I have derived the very constitu- 
tion of the simple elements of matter, presents a system that is midway 
between that of Leibniz & that of Newton ; it has very much in common 
with both, & differs very much from either ; &, as it is immensely more 
simple than either, it is undoubtedly suitable in a marvellous degree for 

deriving all the general properties of bodies, & certain of the special properties also, by 

means of the most rigorous demonstrations. 

2. It indeed holds to those simple & perfectly non-extended primary elements upon what there is in 
which is founded the theory of Leibniz ; & also to the mutual forces, which vary as the * s ^" " { to $^ 
distances of the points from one another vary, the characteristic of the theory of Newton ; ton *& Leibniz. 

in addition, it deals not only with the kind of forces, employed by Newton, which oblige 
the points to approach one another, & are commonly called attractions ; but also it 
considers forces of a kind that engender recession, & are called repulsions. Further, the 
idea is introduced in such a manner that, where attraction ends, there, with a change of 
distance, repulsion begins ; this idea, as a matter of fact, was suggested by Newton in the 
last of his ' Questions on Optics ', & he illustrated it by the example of the passage from 
positive to negative, as used in algebraical formulas. Moreover there is this common point 
between either of the theories of Newton & Leibniz & my own ; namely, that any particle 
of matter is connected with every other particle, no matter how great is the distance 
between them, in such a way that, in accordance with a change in the position, no matter 
how slight, of any one of them, the factors that determine the motions of all the rest are 
altered ; &, unless it happens that they all cancel one another (& this is infinitely impro- 
bable), some motion, due to the change of position in question, will take place in every one 
of them. 

3. But my Theory differs in a marked degree from that of Leibniz. For one thing, How it differs from, 
because it does not admit the continuous extension that arises from the idea of consecutive, 
non-extended points touching one another ; here, the difficulty raised in times gone by in 

opposition to Zeno, & never really or satisfactorily answered (nor can it be answered), with 
regard to compenetration of all kinds with non-extended consecutive points, still holds the 
same force against the system of Leibniz. For another thing, it admits homogeneity 
amongst the elements, all distinction between masses depending on relative position only, 
& different combinations of the elements ; for this homogeneity amongst the elements, & 
the reason for the difference amongst masses, Nature herself provides us with the analogy. 
Chemical operations especially do so ; for, since the result of the analysis of compound 
substances leads to classes of elementary substances that are so comparatively few in num- 
ber, & still less different from one another in nature ; it strongly suggests that, the further 
analysis can be pushed, the greater the simplicity, & homogeneity, that ought to be attained ; 
thus, at length, we should have, as the result of a final decomposition, homogeneity & 
simplicity of the highest degree. Against this homogeneity & simplicity, the principle of 
indiscernibles, & the doctrine of sufficient reason, so long & strongly advocated by the 
followers of Leibniz, can, in my opinion at least, avail in not the slightest degree. 

4. My Theory also differs as widely as possible from that of Newton. For one thing, HOW it differs from, 
because it explains by means of a single law of forces all those things that Newton himself, * surpasses, the 

i i i i. . X f-\ , i i theory of Newton. 

in the last of his Questions on Uptics , endeavoured to explain by the three principles 
of gravity, cohesion & fermentation ; nay, & very many other things as well, which do not 
altogether follow from those three principles. Further, this law is expressed by a single 
algebraical formula, & not by one composed of several formulae compounded together ; or 
by a single continuous geometrical curve. For another thing, it admits forces that at very 
small distances are not positive or attractive, as Newton supposed, but negative or repul- 




infinitum, quo distantise in infinitum decrescant. Unde illud necessario consequitur, ut nee 
cohaesio a contactu immediate oriatur, quam ego quidem longe aliunde desumo ; nee ullus 
immediatus, &, ut ilium appellare soleo, mathematicus materiae contactus habeatur, quod 
simplicitatem, & inextensionem inducit elementorum, quae ipse variarum figurarum voluit, 
& partibus a se invicem distinctis composita, quamvis ita cohasrentia, ut nulla Naturae vi 
dissolvi possit compages, & adhaesio labefactari, quas adhaesio ipsi, respectu virium nobis 
cognitarum, est absolute infinita. 

Ubi de ipsa ctum 5. Quae ad ejusmodi Theoriam pertinentia hucusque sunt edita, continentur disserta- 

ante ; & quid pro- tionibus meis, De viribus vivis, edita Anno 1741;, De Lumine A. 1748, De Leee Continuitatis 

ml<mm " . T r^ ... . . rj . ... 

A. 1754, De Lege virium in natura existentium A. 1755, De divisibihtate materite, C5 principiis 
corporum A. 1757, ac in meis Supplementis Stayanae Philosophiae versibus traditae, cujus primus 
Tomus prodiit A. 1755 : eadem autem satis dilucide proposuit, & amplissimum ipsius per 
omnem Physicam demonstravit usum vir e nostra Societate doctissimus Carolus Benvenutus 
in sua Physics Generalis Synopsi edita Anno 1754. In ea Synopsi proposuit idem & meam 
deductionem aequilibrii binarum massarum, viribus parallelis animatarum, quas ex ipsa mea 
Theoria per notissimam legem compositionis virium, & aequalitatis inter actionem, & reac- 
tionem, fere sponte consequitur, cujus quidem in supplementis illis 4. ad lib. 3. mentionem 
feci, ubi & quae in dissertatione De centra Gravitatis edideram, paucis proposui ; & de centre 
oscillationis agens, protuli aliorum methodos praecipuas quasque, quae ipsius determinationem 
a subsidiariis tantummodo principiis quibusdam repetunt. Ibidem autem de sequilibrii 
centre agens illud affirmavi : In Natura nullce sunt rigidce virgce, infiexiles, & omni gravitate, 
ac inertia carentes, adeoque nee revera ullce leges pro Us conditcz ; & si ad genuina, & simpli- 
cissima natures principia, res exigatur, invenietur, omnia pendere a compositione virium, quibus in 
se invicem agunt particula materice ; a quibus nimirum viribus omnia Natures pb&nomena 
proficiscuntur. Ibidem autem exhibitis aliorum methodis ad centrum oscillationis perti- 
nentibus, promisi, me in quarto ejusdem Philosophiae tomo ex genuinis principiis investiga- 
turum, ut aequilibrii, sic itidem oscillationis centrum. 

Qua occasione hoc 6. Porro cum nuper occasio se mihi praebuisset inquirendi in ipsum oscillationis centrum 
turn 'opus." Cnp ex me i s principiis, urgente Scherffero nostro viro doctissimo, qui in eodem hoc Academico 
Societatis Collegio nostros Mathesim docet ; casu incidi in theorema simplicisimum sane, & 
admodum elegans, quo trium massarum in se mutuo agentium comparantur vires, [4] quod 
quidem ipsa fortasse tanta sua simplicitate effugit hucusque Mechanicorum oculos ; nisi 
forte ne effugerit quidem, sed alicubi jam ab alio quopiam inventum, & editum, me, quod 
admodum facile fieri potest, adhuc latuerit, ex quo theoremate & asquilibrium, ac omne 
vectium genus, & momentorum mensura pro machinis, & oscillationis centrum etiam pro 
casu, quo oscillatio fit in latus in piano ad axem oscillationis perpendiculari, & centrum 
percussionis sponte fluunt, & quod ad sublimiores alias perquisitiones viam aperit admodum 
patentem. Cogitaveram ego quidem initio brevi dissertatiuncula hoc theorema tantummodo 
edere cum consectariis, ac breve Theoriae meae specimen quoddam exponere ; sed paullatim 
excrevit opusculum, ut demum & Theoriam omnem exposuerim ordine suo, & vindicarim, 
& ad Mechanicam prius, turn ad Physicam fere universam applicaverim, ubi & quae maxima 
notatu digna erant, in memoratis dissertationibus ordine suo digessi omnia, & alia adjeci 
quamplurima, quae vel olim animo conceperam, vel modo sese obtulerunt scribenti, & omnem 
hanc rerum farraginem animo pervolventi. 

eiementa in- 7. Prima elementa materiae mihi sunt puncta prorsus indivisibilia, & inextensa, quae in 

i mr ftenso vacuo ita dispersa sunt, ut bina quaevis a se invicem distent per aliquod intervallum, 
quod quidem indefinite augeri potest, & minui, sed penitus evanescere non potest, sine 
conpenetratione ipsorum punctorum : eorum enim contiguitatem nullam admitto possi- 
bilem ; sed illud arbitror omnino certum, si distantia duorum materiae punctorum sit nulla, 
idem prorsus spatii vulgo concept! punctum indivisibile occupari ab utroque debere, & 


sive ; although these also become greater & greater indefinitely, as the distances decrease 
indefinitely. From this it follows of necessity that cohesion is not a consequence of imme- 
diate contact, as I indeed deduce from totally different considerations ; nor is it possible 
to get any immediate or, as I usually term it, mathematical contact between the parts of 
matter. This idea naturally leads to simplicity & non-extension of the elements, such as 
Newton himself postulated for various figures ; & to bodies composed of parts perfectly 
distinct from one another, although bound together so closely that the ties could not be 
broken or the adherence weakened by any force in Nature ; this adherence, as far as the 
forces known to us are concerned, is in his opinion unlimited. 

5. What has already been published relating to this kind of Theory is contained in my when & where I 
dissertations, De Viribus vivis, issued in 1745, De Lumine, 1748, De Lege Continuitatis, * th^theory'* 
1754, De Lege virium in natura existentium, 1755, De divisibilitate materia, y principiis & a promise that i 
corporum, 1757, & in my Supplements to the philosophy of Benedictus Stay, issued in verse, made> 

of which the first volume was published in 1755. The same theory was set forth with 
considerable lucidity, & its extremely wide utility in the matter of the whole of Physics 
was demonstrated, by a learned member of our Society, Carolus Benvenutus, in his Physics 
Generalis Synopsis published in 1754. In this synopsis he also at the same time gave my 
deduction of the equilibrium of a pair of masses actuated by parallel forces, which follows 
quite naturally from my Theory by the well-known law for the composition of forces, & 
the equality between action & reaction ; this I mentioned in those Supplements, section 
4 of book 3, & there also I set forth briefly what I had published in my dissertation De 
centra Gravitatis. Further, dealing with the centre of oscillation, I stated the most note- 
worthy methods of others who sought to derive the determination of this centre from 
merely subsidiary principles. Here also, dealing with the centre of equilibrium, I asserted : 
" In Nature there are no rods that are rigid, inflexible, totally devoid of weight & inertia ; 
y so, neither are there really any laws founded on them. If the matter is worked back to the 
genuine W simplest natural principles, it will be found that everything depends on the com- 
position of the forces with which the particles of matter act upon one another ; y from these 
very forces, as a matter of fact, all phenomena of Nature take their origin." Moreover, here 
too, having stated the methods of others for the determination of the centre of oscillation, 
I promised that, in the fourth volume of the Philosophy, I would investigate by means of 
genuine principles, such as I had used for the centre of equilibrium, the centre of 
oscillation as well. 

6. Now, lately I had occasion to investigate this centre of oscillation, deriving it from The occasion that 
my own principles, at the request of Father Scherffer, a man of much learning, who teaches |^ 
mathematics in this College of the Society. Whilst doing this, I happened to hit upon a matter. 

really most simple & truly elegant theorem, from which the forces with which three 
masses mutually act upon one another are easily to be found ; this theorem, perchance 
owing to its extreme simplicity, has escaped the notice of mechanicians up till now (unless 
indeed perhaps it has not escaped notice, but has at some time previously been discovered 
& published by some other person, though, as may very easily have happened, it may not 
have come to my notice). From this theorem there come, as the natural consequences, 
the equilibrium & all the different kinds of levers, the measurement of moments for 
machines, the centre of oscillation for the case in which the oscillation takes place sideways 
in a plane perpendicular to the axis of oscillation, & also the centre of percussion ; it opens 
up also a beautifully clear road to other and more sublime investigations. Initially, my 
idea was to publish in a short esssay merely this theorem & some deductions from it, & thus 
to give some sort of brief specimen of my Theory. But little by little the essay grew in 
length, until it ended in my setting forth in an orderly manner the whole of the theory, 
giving a demonstration of its truth, & showing its application to Mechanics in the first place, 
and then to almost the whole of Physics. To it I also added not only those matters that 
seemed to me to be more especially worth mention, which had all been already set forth 
in an orderly manner in the dissertations mentioned above, but also a large number of other 
things, some of which had entered my mind previously, whilst others in some sort pb truded 
themselves on my notice as I was writing & turning over in my mind all this conglomer- 
ation of material. 

7. The primary elements of matter are in my opinion perfectly indivisible & non- The primary eie- 
extended points ; they are so scattered in an immense vacuum that every two of them are ^biVnon^xtended 
separated from one another by a definite interval ; this interval can be indefinitely & they are not 
increased or diminished, but can never vanish altogether without compenetration of the c 

points themselves ; for I do not admit as possible any immediate contact between them. 
On the contrary I consider that it is a certainty that, if the distance between two points 
of matter should become absolutely nothing, then the very same indivisible point of space, 
according to the usual idea of it, must be occupied by both together, & we have true 


haberi veram, ac omnimodam conpenetrationem. Quamobrem non vacuum ego quidem 
admitto disseminatum in materia, sed materiam in vacuo disseminatam, atque innatantem. 

Eorum inertias vis g j n n ; sce punctis admitto determinationem perseverandi in eodem statu quietis, vel 

cujusmodi. . r . r . ,. , . . , . J . . i * XT ' 

motus umiormis in directum l) m quo semel sint posita, si seorsum smgula in JNatura 
existant ; vel si alia alibi extant puncta, componendi per notam, & communem metho- 
dum compositionis virium, & motuum, parallelogrammorum ope, praecedentem motum 
cum mo-[5]-tu quern determinant vires mutuae, quas inter bina quaevis puncta agnosco 
a distantiis pendentes, & iis mutatis mutatas, juxta generalem quandam omnibus com- 
munem legem. In ea determinatione stat ilia, quam dicimus, inertiae vis, quae, an a 
libera pendeat Supremi Conditoris lege, an ab ipsa punctorum natura, an ab aliquo iis 
adjecto, quodcunque, istud sit, ego quidem non quaere ; nee vero, si velim quasrere, in- 
veniendi spem habeo ; quod idem sane censeo de ea virium lege, ad quam gradum jam facio. 

Eorundem vires g Censeo igitur bina quaecunque materiae puncta determinari asque in aliis distantiis 

mutuae in alus , y ,.. , - 1 . . . 

distantiis attrac- ad mutuum accessum, in alns ad recessum mutuum, quam ipsam determinationem appello 
tivae, in aliis re- v im, in priore casu attractivam, in posteriore repulsivam, eo nomine non agendi modum, sed 

pulsivae : v i n u m . , r . . . , '. . . 

ejusmodi exempia. ipsam determinationem expnmens, undecunque provemat, cujus vero magnitude mutatis 
distantiis mutetur & ipsa secundum certam legem quandam, quae per geometricam lineam 
curvam, vel algebraicam formulam exponi possit, & oculis ipsis, uti moris est apud Mechanicos 
repraesentari. Vis mutuae a distantia pendentis, & ea variata itidem variatae, atque ad omnes 
in immensum & magnas, & parvas distantias pertinentis, habemus exemplum in ipsa 
Newtoniana generali gravitate mutata in ratione reciproca duplicata distantiarum, qua; 
idcirco numquam e positiva in negativam migrare potest, adeoque ab attractiva ad repul- 
sivam, sive a determinatione ad accessum ad determinationem ad recessum nusquam migrat. 
Verum in elastris inflexis habemus etiam imaginem ejusmodi vis mutuae variatae secundum 
distantias, & a determinatione ad recessum migrantis in determinationem ad accessum, & 
vice versa. Ibi enim si duae cuspides, compresso elastro, ad se invicem accedant, acquirunt 
determinationem ad recessum, eo majorem, quo magis, compresso elastro, distantia 
decrescit ; aucta distantia cuspidum, vis ad recessum minuitur, donee in quadam distantia 
evanescat, & fiat prorsus nulla ; turn distantia adhuc aucta, incipit determinatio ad accessum, 
quae perpetuo eo magis crescit, quo magis cuspides a se invicem recedunt : ac si e contrario 
cuspidum distantia minuatur perpetuo ; determinatio ad accessum itidem minuetur, 
evanescet, & in determinationem ad recessum mutabitur. Ea determinatio oritur utique 
non ab immediata cuspidum actione in se invicem, sed a natura, & forma totius intermediae 
laminae plicatae ; sed hie physicam rei causam non merer, & solum persequor exemplum 
determinationis ad accessum, & recessum, quae determinatio in aliis distantiis alium habeat 
nisum, & migret etiam ab altera in alteram. 

virium earundero 10. Lex autem virium est ejusmodi, ut in minimis distantiis sint repulsivae, atque eo 

majores in infmitum, quo distantiae ipsae minuuntur in infinitum, ita, ut pares sint extinguen- 
[6]-dae cuivis velocitati utcunque magnae, cum qua punctum alterum ad alterum possit 
accedere, antequam eorum distantia evanescat ; distantiis vero auctis minuuntur ita, ut in 
quadam distantia perquam exigua evadat vis nulla : turn adhuc, aucta distantia, mutentur in 
attractivas, prime quidem crescentes, turn decrescentes, evanescentes, abeuntes in repulsivas, 
eodem pacto crescentes, deinde decrescentes, evanescentes, migrantes iterum in attractivas, 
atque id per vices in distantiis plurimis, sed adhuc perquam exiguis, donee, ubi ad aliquanto 
majores distantias ventum sit, incipiant esse perpetuo attractivae, & ad sensum reciproce 

(a) Id quidem respectu ejus spatii, in quo continemur nos, W omnia quis nostris observari sensibus possunt, corpora ; 
quod quiddam spatium si quiescat, nihil ego in ea re a reliquis differo ; si forte moveatur motu quopiam, quern motum 
ex hujusmodi determinatione sequi debeant ipsa materia puncta ; turn bcec mea erit quiedam non absoluta, sed respectiva 
inertia: vis, quam ego quidem exposui W in dissertatione De Maris aestu fcf in Supplementis Stayanis Lib. I. 13 ; 
ubi etiam illud occurrit, quam oh causam ejusmodi respectivam inertiam excogitarim, & quibus rationihus evinci putem, 
absolutam omnino demonstrari non posse ; sed ea hue non pertinent. 


compenetration in every way. Therefore indeed I do not admit the idea of vacuum 
interspersed amongst matter, but I consider that matter is interspersed in a vacuum & 
floats in it. 

8. As an attribute of these points I admit an inherent propensity to remain in the The nat . ure ? f the 
same state of rest, or of uniform motion in a straight line, () in which they are initially the" possess. 1 * 
set, if each exists by itself in Nature. But if there are also other points anywhere, there 

is an inherent propensity to compound (according to the usual well-known composition of 
forces & motions by the parallelogram law), the preceding motion with the motion which 
is determined by the mutual forces that I admit to act between any two of them, depending 
on the distances & changing, as the distances change, according to a certain law common 
to them all. This propensity is the origin of what we call the ' force of inertia ' ; whether 
this is dependent upon an arbitrary law of the Supreme Architect, or on the nature of points 
itself, or on some attribute of them, whatever it may be, I do not seek to know ; even if I 
did wish to do so, I see no hope of finding the answer ; and I truly think that this also 
applies to the law of forces, to which I now pass on. 

9. I therefore consider that any two points of matter are subject to a determination The mutual forces 
to approach one another at some distances, & in an equal degree recede from one another at Stw^*^!* 
other distances. This determination I call ' force ' ; in the first case ' attractive ', in the distances & repui- 
second case ' repulsive ' ; this term does not denote the mode of action, but the propen- ^mpies 

sity itself, whatever its origin, of which the magnitude changes as the distances change ; this kind, 
this is in accordance with a certain definite law, which can be represented by a geometrical 
curve or by an algebraical formula, & visualized in the manner customary with Mechanicians. 
We have an example of a force dependent on distance, & varying with varying distance, & 
pertaining to all distances either great or small, throughout the vastness of space, in the 
Newtonian idea of general gravitation that changes according to the inverse squares of the 
distances : this, on account of the law governing it, can never pass from positive to nega- 
tive ; & thus on no occasion does it pass from being attractive to being repulsive, i.e., from 
a propensity to approach to a propensity to recession. Further, in bent springs we have 
an illustration of that kind of mutual force that varies according as the distance varies, & 
passes from a propensity to recession to a propensity to approach, and vice versa. For 
here, if the two ends of the spring approach one another on compressing the spring, they 
acquire a propensity for recession that is the greater, the more the distance diminishes 
between them as the spring is compressed. But, if the distance between the ends is 
increased, the force of recession is diminished, until at a certain distance it vanishes and 
becomes absolutely nothing. Then, if the distance is still further increased, there begins a 
propensity to approach, which increases more & more as the ends recede further & further 
away from one another. If now, on the contrary, the distance between the ends is con- 
tinually diminished, the propensity to approach also diminishes, vanishes, & becomes changed 
into a propensity to recession. This propensity certainly does not arise from the imme- 
diate action of the ends upon one another, but from the nature & form of the whole of the 
folded plate of metal intervening. But I do not delay over the physical cause of the thing 
at this juncture ; I only describe it as an example of a propensity to approach & recession, 
this propensity being characterized by one endeavour at some distances & another at other 
distances, & changing from one propensity to another. 

10. Now the law of forces is of this kind ; the forces are repulsive at very small dis- The Iaw . of forces 
tances, & become indefinitely greater & greater, as the distances are diminished indefinitely, for the pomts- 

in such a manner that they are capable of destroying any velocity, no matter how large it 
may be, with which one point may approach another, before ever the distance between 
them vanishes. When the distance between them is increased, they are diminished in such 
a way that at a certain distance, which is extremely small, the force becomes nothing. 
Then as the distance is still further increased, the forces are change-d to attractive forces ; 
these at first increase, then diminish, vanish, & become repulsive forces, which in the same 
way first increase, then diminish, vanish, & become once more attractive ; & so on, in turn, 
for a very great number of distances, which 1 are all still very^ minute : until, finally, when 
we get to comparatively great distances, they begin to be continually attractive & approxi- 

(a) This indeed holds true for that space in which we, and all bodies that can be observed by our senses, are 
contained. Now, if this space is at rest, I do not differ from other philosophers with regard to the matter in question ; 
but if perchance space itself moves in some way or other, what motion ought these points of matter to comply with owing 
to this kind of propensity ? In that case Ms force of inertia that I postulate is not absolute, but relative ; as indeed 
I explained both in the dissertation De Maris Aestu, and also in the Supplements to Stay's Philosophy, book I, section 
13. Here also will be found the conclusions at which I arrived with regard to relative inertia of this sort, and the 
arguments by which I think it is proved that it is impossible to show that it is generally abxlute. But these things do 
not concern us at present. 



proportionales quadratis distantiarum, atque id vel utcunque augeantur distantiae etiam in 
infinitum, vel saltern donee ad distantias deveniatur omnibus Planetarum, & Cometarum 
distantiis longe majores. 

Leg is simpiicitas ii. Hujusmodi lex primo aspectu videtur admodum complicata, & ex diversis legibus 

exprimibihs per temere j nter se coagmentatis coalescens ; at simplicissima, & prorsus incomposita esse potest, 

COIlLlIlUtllTl CUf VclIUi t i 1*1* A 1 1 " J" 1 

expressa videlicet per unicam contmuam curvam, vel simphcem Algebraicam iormulam, uti 
innui superius. Hujusmodi curva linea est admodum apta ad sistendam oculis ipsis ejusmodi 
legem, nee requirit Geometram, ut id praestare possit : satis est, ut quis earn intueatur 
tantummodo, & in ipsa ut in imagine quadam solemus intueri depictas res qualescunque, 
virium illarum indolem contempletur. In ejusmodi curva eae, quas Geometrae abscissas 
dicunt, & sunt segmenta axis, ad quern ipsa refertur curva, exprimunt distantias binorum 
punctorum a se invicem : illae vero, quae dicuntur ordinatae, ac sunt perpendiculares lineee 
ab axe ad curvam ductae, referunt vires : quae quidem, ubi ad alteram jacent axis partem, 
exhibent vires attractivas ; ubi jacent ad alteram, rcpulsivas, & prout curva accedit ad axem, 
vel recedit, minuuntur ipsae etiam, vel augentur : ubi curva axem secat, & ab altera ejus 
parte transit ad alteram, mutantibus directionem ordinatis, abeunt ex positivis in negativas, 
vel vice versa : ubi autem arcus curvae aliquis ad rectam quampiam axi perpendicularem 
in infinitum productam semper magis accedit ita ultra quoscumque limites, ut nunquam in 
earn recidat, quern arcum asymptoticum appellant Geometrae, ibi vires ipsae in infinitum 

Forma curvae ips- 

12. Ejusmodi curvam exhibui, & exposui in dissertationibus De viribus vivis a Num. 51, 
De Lumine Num. 5, De Lege virium in Naturam existentium a Num. 68, & in sua Synopsi 
Physics Generalis P. Benvenutus eandem protulit a Num. 108. En brevem quandemejus 
ideam. In Fig. i, Axis C'AC habet in puncto A asymptotum curvae rectilineam AB 
indefinitam, circa quam habentur bini curvae rami hinc, & inde aequales, prorsus inter se, & 
similes, quorum alter DEFGHIKLMNOPQRSTV habet inprimis arcum ED [7] asympto- 
ticum, qui nimirum ad partes BD, si indefinite producatur ultra quoscunque limites, semper 
magis accedit ad rectam AB productam ultra quoscunque limites, quin unquam ad eandem 
deveniat ; hinc vero versus DE perpetuo recidit ab eadam recta, immo etiam perpetuo 
versus V ab eadem recedunt arcus reliqui omnes, quin uspiam recessus mutetur in accessum. 
Ad axem C'C perpetuo primum accedit, donee ad ipsum deveniat alicubi in E ; turn eodem 
ibi secto progreditur, & ab ipso perpetuo recedit usque ad quandam distantiam F, postquam 
recessum in accessum mutat, & iterum ipsum axem secat in G, ac flexibus continuis contor- 
quetur circa ipsum, quern pariter secat in punctis quamplurimis, sed paucas admodum 
ejusmodi sectiones figura exhibet, uti I, L, N, P, R. Demum is arcus desinit in alterum 
crus TpsV, jacens ex parte opposita axis respectu primi cruris, quod alterum crus ipsum 
habet axem pro asymptoto, & ad ipsum accedit ad sensum ita, ut distantiae ab ipso sint in 
ratione reciproca duplicata distantiarum a recta BA. 

Abscissae exprimen- 

d!nate Sta exprimen- 
tes vires. 

13. Si ex quovis axis puncto a, b, d, erigatur usque ad curvam recta ipsi perpendicularis 
a S> ^ r ' ^h , segmentum axis Aa, Ab, Ad, dicitur abscissa, & refert distantiam duorum materiae 
punctorum quorumcunque a se invicem ; perpendicularis ag, br, db , dicitur ordinata, & 
exhibet vim repulsivam, vel attractivam, prout jacet respectu axis ad partes D, vel oppositas. 

Mutationes ordina- 
tarum, & virium iis 

14. Patet autem, in ea curvae forma ordinatam ag augeri ultra quoscunque limites, si 
abscissa Aa, minuatur pariter ultra quoscunque limites ; quae si augeatur, ut abeat in Ab, 
ordinata minuetur, & abibit in br, perpetuo imminutam in accessu b ad E, ubi evanescet : 
turn aucta abscissa in Ad, mutabit ordinata directionem in dh , ac ex parte opposita augebitur 
prius usque ad F, turn decrescet per il usque ad G, ubi evanescet, & iterum mutabit 
directionem regressa in mn ad illam priorem, donee post evanescentiam, & directionis 
mutationem factam in omnibus sectionibus I, L, N, P, R, fiant ordinatas op, vs, directionis 
constantis, & decrescentes ad sensum in ratione reciproca duplicata abscissarum Ao, Av. 
Quamobrem illud est manifestum, per ejusmodi curvam exprimi eas ipsas vires, initio 





mately inversely proportional to the squares of the distances. This holds good as the 
distances are increased indefinitely to any extent, or at any rate until we get to distances 
that are far greater than all the distances of the planets & comets. 

11. A law of this kind will seem at first sight to be very complicated, & to be the result The simplicity of 

of combining together several different laws in a haphazard sort of way ; but it can be of the law can ^ re ~ 
^.t. i 1 i j o v j i v i 1 i r presented by means 

the simplest kind & not complicated in the slightest degree ; it can be represented for of a continuous 
instance by a single continuous curve, or by an algebraical formula, as I intimated above. curve - 
A curve of this sort is perfectly adapted to the .graphical representation of this sort of law, 
& it does not require a knowledge of geometry to set it forth. It is sufficient for anyone 
merely to glance at it, & in it, just as in a picture we are accustomed to view all manner of 
things depicted, so will he perceive the nature of these forces. In a curve of this kind, 
those lines, that geometricians call abscissae, namely, segments of the axis to which the 
curve is referred, represent the distances of two points from one another ; & those, which 
we called ordinates, namely, lines drawn perpendicular to the axis to meet the curve, repre- 
sent forces. These, when they lie on one side of the axis represent attractive forces, and, 
when they lie on the other side, repulsive forces ; & according as the curve approaches the 
axis or recedes from it, they too are diminished or increased. When the curve cuts the 
axis & passes from one side of it to the other, the direction of the ordinates being changed 
in consequence, the forces pass from positive to negative or vice versa. When any arc of 
the curve approaches ever more closely to some straight line perpendicular to the axis and 
indefinitely produced, in such a manner that, even if this goes on beyond all limits, yet 
the curve never quite reaches the line (such an arc is called asymptotic by geometricians), 
then the forces themselves will increase indefinitely. 

12. I set forth and explained a curve of this sort in my dissertations De Firibus vivis The form of the 
(Art. 51), De Lumine (Art. 5), De lege virium in Natura existentium (Art. 68) ; and Father curve - 
Benvenutus published the same thing in his Synopsis Physicce Generalis (Art. 108). This 

will give you some idea of its nature in a few words. 

In Fig. i the axis C'AC has at the point A a straight line AB perpendicular to itself, 
which is an asymptote to the curve ; there are two branches of the curve, one on each side 
of AB, which are equal & similar to one another in every way. Of these, one, namely 
DEFGHIKLMNOPQRSTV, has first of all an asymptotic arc ED ; this indeed, if it is 
produced ever so far in the direction ED, will approach nearer & nearer to the straight line 
AB when it also is produced indefinitely, but will never reach it ; then, in the direction 
DE, it will continually recede from this straight line, & so indeed will all the rest of the arcs 
continually recede from this straight line towards V. The first arc continually approaches 
the axis C'C, until it meets it in some point E ; then it cuts it at this point & passes on, 
continually receding from the axis until it arrives at a certain distance given by the point 
F ; after that the recession changes to an approach, & it cuts the axis once more in G ; & 
so on, with successive changes of curvature, the curve winds about the axis, & at the same 
time cuts it in a number of points that is really large, although only a very few of the 
intersections of this kind, as I, L, N, P, R, are shown in the diagram. Finally the arc of the 
curve ends up with the other branch TpsV, lying on the opposite side of the axis with 
respect to the first branch ; and this second branch has the axis itself as its asymptote, 
& approaches it approximately in such a manner that the distances from the axis are in 
the inverse ratio of the squares of the distances from the straight line AB. 

13. If from any point of the axis, such as a, b, or d, there is erected a straight line per- The abscissae re- 
pendicular to it to meet the curve, such as ag, br, or db then the segment of the axis, Aa, res ^Jg 

Ab, or Ad, is called the abscissa, & represents the distance of any two points of matter from forces, 
one another ; the perpendicular, ag, br, or dh, is called the ordinate, & this represents the 
force, which is repulsive or attractive, according as the ordinate lies with regard to the 
axis on the side towards D, or on the opposite side. 

14. Now it is clear that, in a curve of this form, the ordinate ag will be increased Change in the or- 
beyond all bounds, if the abscissa Aa is in the same way diminished beyond all bounds ; & fbat tlfey reprSent! 
if the latter is increased and becomes Ab, the ordinate will be diminished, & it will become 

br, which will continually diminish as b approaches to E, at which point it will vanish. 
Then the abscissa being increased until it becomes Ad, the ordinate will change its direction 
as it becomes db, & will be increased in the opposite direction at first, until the point F is 
reached, when it will be decreased through the value il until the point G is attained, at 
which point it vanishes ; at the point G, the ordinate will once more change its direction 
as it returns to the position mn on the same side of the axis as at the start. Finally, after 
vanishing & changing direction at all points of intersection with the axis, such as I, L, N, 
P, R, the ordinates take the several positions indicated by op, vs : here the direction remains 
unchanged, & the ordinates decrease approximately in the inverse ratio of the squares of 
the abscissae Ao, Av. Hence it is perfectly evident that, by a curve of this kind, we can 



Discrimen hu us 
legis virium a 
gravitate N e w- 
toniana : ejus usus 
in Physica : ordo 

Occasio inveniendae 
Theories ex consid- 
eraticine impulsus. 


repulsivas, & imminutis in infinitum distantiis auctas in infinitum, auctis imminutas, turn 
evanescentes, abeuntes, mutata directione, in attractivas, ac iterum evenescentes, mutatasque 
per vices : donee demum in satis magna distantia evadant attractive ad sensum in ratione 
reciproca duplicata distantiarum. 

15. Haec virium lex a Newtoniana gravitate differt in ductu, & progressu curvae earn 
exprimentis quse nimirum, ut in fig. 2, apud Newtonum est hyperbola DV gradus tertii, 
jacens tota citra axem, quern nuspiam 

secat, jacentibus omni-[8]-bus ordinatis 
vs, op, bt, ag ex parte attractiva, ut 
idcirco nulla habeatur mutatio e positivo 
ad negativum, ex attractione in repulsi- 
onem, vel vice versa ; caeterum utraque 
per ductum exponitur curvae continue 
habentis duo crura infinita asymptotica 
in ramis singulis utrinque in infinitum 
productis. Ex hujusmodi autem virium 
lege, & ex solis principiis Mechanicis 
notissimis, nimirum quod ex pluribus 
viribus, vel motibus componatur vis, vel 
motus quidam ope parallelogrammorum, 
quorum latera exprimant vires, vel mo- 
tus componentes, & quod vires ejusmodi 

in punctis singulis, tempusculis singulis aequalibus, inducant velocitates, vel motus proportion- 
ales sibi, omnes mihi profluunt generales, & praecipuae quacque particulars proprietates cor- 
porum,uti etiam superius innui, nee ad singulares proprietates derivandas in genere afHrmo, eas 
haberi per diversam combinationem, sed combinationes ipsas evolvo, & geometrice demon- 
stro, quae e quibus combinationibus phasnomena, & corporum species oriri debeant. Verum 
antequam ea evolvo in parte secunda, & tertia, ostendam in hac prima, qua via, & quibus 
positivis rationibus ad earn virium legem devenerim, & qua ratione illam elementorum 
materiae simplicitatem eruerim, turn quas difHcultatem aliquam videantur habere posse, 

1 6. Cum anno 1745 De Viribus vivis dissertationem conscriberem, & omnia, quse a 
viribus vivis repetunt, qui Leibnitianam tuentur sententiam, & vero etiam plerique ex iis, 
qui per solam velocitatem vires vivas metiuntur, repeterem immediate a sola velocitate 
genita per potentiarum vires, quae juxta communem omnium Mechanicorum sententiam 
velocitates vel generant, vel utcunque inducunt proportionales sibi, & tempusculis, quibus 
agunt, uti est gravitas, elasticitas, atque aliae vires ejusmodi ; ccepi aliquant: o diligentius 
inquirere in earn productionem velocitatis, quae per impulsum censetur fieri, ubi tota 
velocitas momento temporis produci creditur ab iis, qui idcirco percussionis vim infinities 
majorem esse censent viribus omnibus, quae pressionem solam momentis singulis exercent. 
Statim illud mihi sese obtulit, alias pro percussionibus ejusmodi, quee nimirum momento 
temporis finitam velocitatem inducant, actionum leges haberi debere. 


origo ejusdem ex 17. Verum re altius considerata, mihi illud incidit, si recta utamur ratiocinandi methodo, 

susTmrnedUatTalin eum agendi modum submovendum esse a Natura, quae nimirum eandem ubique virium 

lege Continuitatis. legem, ac eandem agendi rationem adhibeat : impulsum nimirum immediatum alterius 

corporis in alterum, & immediatam percussionem haberi non posse sine ilia productione 

finitse velocitatis facta momento temporis indivisibili, & hanc sine saltu quodam, & Isesione 

illius, quam legem Continuitatis appellant, quam quidem legem in Natura existere, & quidem 

satis [9] valida ratione evinci posse existimabam. En autem ratiocinationem ipsam, qua 

turn quidem primo sum usus, ac deinde novis aliis, atque aliis meditationibus illustravi, ac 


minus velox. 

Laesio legis Continu- 18. Concipiantur duo corpora aequalia, quae moveantur in directum versus eandem 

cOTpus^efocruTim- pl a g am > & id, quod praecedit, habeat gradus velocitatis 6, id vero, quod ipsum persequitur 
mediate incurrat in gradus 12. Si hoc posterius cum sua ilia velocitate illaesa deveniat ad immediatum contactum 
cum illo priore ; oportebit utique, ut ipso momento temporis, quo ad contactum devenerint, 
illud posterius minuat velocitatem suam, & illud primus suam augeat, utrumque per saltum, 
abeunte hoc a 12 ad 9, illo a 6 ad 9, sine ullo transitu per intermedios gradus n, & 7 ; 10, & 
8 ; 9^, & 8i, &c. Neque enim fieri potest, ut per aliquam utcunque exiguam continui 


represent the forces in question, which are initially repulsive & increase indefinitely as the 
distances are diminished indefinitely, but which, as the distances increase, are first of all 
diminished, then vanish, then become changed in direction & so attractive, again vanish, 
& change their direction, & so on alternately ; until at length, at a distance comparatively 
great they finally become attractive & are sensibly proportional to the inverse squares of 
the distance. 

ic. This law of forces differs from the law of gravitation enunciated by Newton in Difference between 

. J -nii r i i i i this ' aw f forces 

the construction & development or the curve that represents it ; thus, the curve given in & Newton's law of 
Fie. 2, which is that according to Newton, is DV, a hyperbola of the third degree, lying gravitation ; i t s 

ii i r i i i i nil' 6 use ln Physics ; 

altogether on one side of the axis, which it does not cut at any point ; all the ordmates, the order in which 
such as vs, op, bt, ag lie on the side of the axis representing attractive forces, & there- ^ e t s ^ ects are to 
fore there is no change from positive to negative, i.e., from attraction to repulsion, or 
vice versa. On the other hand, each of the laws is represented by the construction of a 
continuous curve possessing two infinite asymptotic branches in each of its members, if 
produced to infinity on both sides. Now, from a law of forces of this kind, & with the 
help of well-known mechanical principles only, such as that a force or motion can be com- 
pounded from several forces or motions by the help of parallelograms whose sides represent 
the component forces or motions, or that the forces of this kind, acting on single points 
for single small equal intervals of time, produce in them velocities that are proportional to 
themselves ; from these alone, I say, there have burst forth on me in a regular flood all 
the general & some of the most important particular properties of bodies, as I intimated 
above. Nor, indeed, for the purpose of deriving special properties, do I assert that they 
ought to be obtained owing to some special combination of points ; on the contrary I 
consider the combinations themselves, & prove geometrically what phenomena, or what 
species of bodies, ought to arise from this or that combination. Of course, before I 
come to consider, both in the second part and in the third, all the matters mentioned 
above, I will show in this first part in what way, & by what direct reasoning, I have arrived 
at this law of forces, & by what argument I have made out the simplicity of the elements 
of matter ; then I will give an explanation of every point that may seem to present any 
possible difficulty. 

16. In the year 1745, I was putting together my dissertation De Firibus vivis, & had The occasion that 
derived everything that they who adhere to the idea of Leibniz, & the greater number of o^my^L^Trom 
those who measure ' living forces ' by means of velocity only, derive from these ' living the consideration 
forces ' ; as, I say I had derived everything directly & solely from the velocity generated by of im P ulsiv e action, 
the forces of those influences, which, according to the generally accepted view taken by 

all Mechanicians, either generate, or in some way induce, velocities that are proportional 
to themselves & the intervals of time during which they act ; take, for instance, gravity, 
elasticity, & other forces of the same kind. I then began to investigate somewhat more 
carefully that production of velocity which is thought to arise through impulsive action, 
in which the whole of the velocity is credited with being produced in an instant of time by 
those, who think, because of that, that the force of percussion is infinitely greater than all 
forces which merely exercise pressure for single instants. It immediately forced itself upon 
me that, for percussions of this kind, which really induce a finite velocity in an instant of 
time, laws for their actions must be obtained different from the rest. 

17. However, when I considered the matter more thoroughly, it struck me that, if The cause of 
we employ a straightforward method of argument, such a mode of action must be with- w ^ s the^pposftion 
drawn from Nature, which in every case adheres to one & the same law of forces, & the raised to the Law 
same mode of action. I came to the conclusion that really immediate impulsive action of he idea' 

one body on another, & immediate percussion, could not be obtained, without the pro- impulse, 
duction of a finite velocity taking place in an indivisible instant of time, & this would have 
to be accomplished without any sudden change or violation of what is called the Law of 
Continuity ; this law indeed I considered as existing in Nature, & that this could be shown 
to be so by a sufficiently valid argument. The following is the line of argument that I 
employed initially ; afterwards I made it clearer & confirmed it by further arguments & 
fresh reflection. 

1 8. Suppose there are two equal bodies, moving in the same straight line & in the violation of the 
same direction ; & let the one that is in front have a degree of velocity represented by ^ tod^movrng 1 
6, & the one behind a degree represented by 12. If the latter, i.e., the body that was be- more swiftly comes 
hind, should ever reach with its velocity undiminished, & come into absolute contact with, J"* with^another 
the former body which was in front, then in every case it would be necessary that, at the body moving more 
very instant of time at which this contact happened, the hindermost body should diminish slowlv - 

its velocity, & the foremost body increase its velocity, in each case by a sudden change : 
one of them would pass from 12 to 9, the other from 6 to 9, without any passage through 
the intermediate degrees, n & 7, 10 & 8, 9$ & 8f, & so on. For it cannot possibly happen 


temporis particulam ejusmodi mutatio fiat per intermedios gradus, durante contactu. Si 
enim aliquando alterum corpus jam habuit 7 gradus velocitatis, & alterum adhuc retinet 
1 1 ; toto illo tempusculo, quod effluxit ab initio contactus, quando velocitates erant 12, & 6, 
ad id tempus, quo sunt n, & 7, corpus secundum debuit moveri cum velocitate majore, 
quam primum, adeoque plus percurrere spatii, quam illud, proinde anterior ejus superficies 
debuit transcurrere ultra illius posteriorem superficiem, & idcirco pars aliqua corporis 
sequentis cum aliqua antecedentis corporis parte compenetrari debuit, quod cum ob 
impenetrabilitatem, quam in materia agnoscunt passim omnes Physici, & quam ipsi tri- 
buendam omnino esse, facile evincitur, fieri omnino non possit ; oportuit sane, in ipso 
primo initio contactus, in ipso indivisibili momento temporis, quod inter tempus continuum 
praecedens contactum, & subsequens, est indivisibilis limes, ut punctum apud Geometras 
est limes indivisibilis inter duo continue lineae segmenta, mutatio velocitatum facta fuerit 
per saltum sine transitu per intermedias, laesa penitus ilia continuitatis lege, quae itum ab 
una magnitudine ad aliam sine transitu per intermedias omnino vetat. Quod autem in 
corporibus aequalibus diximus de transitu immediato utriusque ad 9 gradus velocitatis, 
recurrit utique in iisdem, vel in utcunque inaequalibus de quovis alio transitu ad numeros 
quosvis. Nimirum ille posterioris corporis excessus graduum 6 momento temporis auferri 
debet, sive imminuta velocitate in ipso, sive aucta in priore, vel in altero imminuta utcunque, 
& aucta in altero, quod utique sine saltu, qui omissis infinitis intermediis velocitatibus 
habeatur, obtineri omnino non poterit. 

Objectio petita a ig. Sunt, qui difficultatem omnem submoveri posse censeant, dicendo, id quidem ita se 

cofporum. dl ' habere debere, si corpora dura habeantur, quae nimirum nullam compressionem sentiant, 
nullam mutationem figurae ; & quoniam hsec a multis excluduntur penitus a Natura ; dum 
se duo globi contingunt, introcessione, [10] & compressione partium fieri posse, ut in ipsis 
corporibus velocitas immutetur per omnes intermedios gradus transitu facto, & omnis 
argumenti vis eludatur. 

Ea uti non posse, 2O fa mpr j m i s e a responsione uti non possunt, quicunque cum Newtono, & vero etiam 

qui admittunt ele- _, \ . . r . j o 

menta soiida, & cum plerisquc veterum Pnilosopnorum pnma elementa matenae omnino dura admittunt, & 

dura - soiida, cum adhaesione infinita, & impossibilitate absoluta mutationis figurae. Nam in primis 

elementis illis solidis, & duris, quae in anteriore adsunt sequentis corporis parte, & in praece- 

dentis posteriore, quae nimirum se mutuo immediate contingunt, redit omnis argumenti vis 

prorsus illaesa. 

Extensionem con- 2 i. Deinde vero illud omnino intelligi sane non potest, quo pacto corpora omnia partes 

primoT pores, 1 *! aliquas postremas circa superficiem non habeant penitus solidas, quae idcirco comprimi 
parietes soiidos, ac ornn i no non possint. In materia quidem, si continua sit, divisibilitas in infinitum haberi 
potest, & vero etiam debet ; at actualis divisio in infinitum difficultates secum trahit sane 
inextricablies ; qua tamen divisione in infinitum ii indigent, qui nullam in corporibus 
admittunt particulam utcunque exiguam compressionis omnis expertem penitus, atque 
incapacem. Ii enim debent admittere, particulam quamcunque actu interpositis poris 
distinctam, divisamque in plures pororum ipsorum velut parietes, poris tamen ipsis iterum 
distinctos. Illud sane intelligi non potest, qui fiat, ut, ubi e vacuo spatio transitur ad corpus, 
non aliquis continuus haberi debeat alicujus in se determinatae crassitudinis paries usque ad 
primum porum, poris utique carens ; vel quomodo, quod eodem recidit, nullus sit extimus, 
& superficiei externae omnium proximus porus, qui nimirum si sit aliquis, parietem habeat 
utique poris expertem, & compressionis incapacem, in quo omnis argumenti superioris vis 
redit prorsus illaesa. 

legis Con- 22. At ea etiam, utcunque penitus inintelligibili, sententia admissa, redit omnis eadem 

iprimis su^r argument! vis in ipsa prima, & ultima corporum se immediate contingentium superficie, vel 
debus, vel punctis. s { nullae continuae superficies congruant, in lineis, vel punctis. Quidquid enim sit id, in quo 
contactus fiat, debet utique esse aliquid, quod nimirum impenetrabilitati occasionem 
praestet, & cogat motum in sequente corpore minui, in prascedente augeri ; id, quidquid est, 
in quo exeritur impenetratibilitatis vis, quo fit immediatus contactus, id sane velocitatem 
mutare debet per saltum, sine transitu per intermedia, & in eo continuitatis lex abrumpi 


that this kind of change is made by intermediate stages in some finite part, however small, 
of continuous time, whilst the bodies remain in contact. For if at any time the one 
body then had 7 degrees of velocity, the other would still retain 1 1 degrees ; thus, during 
the whole time that has passed since the beginning of contact, when the velocities were 
respectively 12 Si 6, until the time at which they are 1 1 & 7, the second body must be moved 
with a greater velocity than the first ; hence it must traverse a greater distance in space 
than the other. It follows that the front surface of the second body must have passed 
beyond the back surface of the first body ; & therefore some part of the body that follows 
behind must be penetrated by some part of the body that goes in front. Now, on account 
of impenetrability, which all Physicists in all quarters recognize in matter, & which can be 
easily proved to be rightly attributed to it, this cannot possibly happen. There really 
must be, in the commencement of contact, in that indivisible instant of time which is an 
indivisible limit between the continuous time that preceded the contact & that subsequent 
to it (just in the same way as a point in geometry is an indivisible limit between two seg- 
ments of a continuous line), a change of velocity taking place suddenly, without any passage 
through intermediate stages ; & this violates the Law of Continuity, which absolutely 
denies the possibility of a passage from one magnitude to another without passing through 
intermediate stages. Now what has been said in the case of equal bodies concerning the 
direct passing of both to 9 degrees of velocity, in every case holds good for such equal bodies, 
or for bodies that are unequal in any way, concerning any other passage to any numbers. 
In fact, the excess of velocity in the hindmost body, amounting to 6 degrees, has to be got 
rid of in an instant of time, whether by diminishing the velocity of this body, or by increasing 
the velocity of the other, or by diminishing somehow the velocity of the one & increasing 
that of the other ; & this cannot possibly be done in any case, without the sudden change 
that is obtained by omitting the infinite number of intermediate velocities. 

19. There are some people, who think that the whole difficulty can be removed by An objection de- 
saying that this is just as it should be, if hard bodies, such as indeed experience no com- ^ ed e x r ^nce nyil o1 
pression or alteration of shape, are dealt with ; whereas by many philosophers hard bodies hard bodies. 

are altogether excluded from Nature ; & therefore, so long as two spheres touch one 
another, it is possible, by introcession & compression of their parts, for it to happen that in 
these bodies the velocity is changed, the passage being made through all intermediate stages ; 
& thus the whole force of the argument will be evaded. 

20. Now in the first place, this reply can not be used by anyone who, following New- This re P'y cannot 
ton, & indeed many of the ancient philosophers as well, admit the primary elements of ^"admit^oiid * 
matter to be absolutely hard & solid, possessing infinite adhesion & a definite shape that it hard elements. 

is perfectly impossible to alter. For the whole force of my argument then applies quite 
unimpaired to those solid and hard primary elements that are in the anterior part of the 
body that is behind, & in the hindmost part of the body that is in front ; & certainly these 
parts touch one another immediately. 

21. Next it is truly impossible to understand in the slightest degree how all bodies do Continuous exten- 
not have some of their last parts just near to the surface perfectly solid, & on that account mary ^resT* walls 
altogether incapable of being compressed. If matter is continuous, it may & must be sub- bounding them, 
ject to infinite divisibility ; but actual division carried on indefinitely brings in its train 

difficulties that are truly inextricable ; however, this infinite division is required by those 
who do not admit that there are any particles, no matter how small, in bodies that are 
perfectly free from, & incapable of, compression. For they must admit the idea that every 
particle is marked off & divided up, by the action of interspersed pores, into many boundary 
walls, so to speak, for these pores ; & these walls again are distinct from the pores them- 
selves. It is quite impossible to understand why it comes about that, in passing from 
empty vacuum to solid matter, we are not then bound to encounter some continuous wall of 
some definite inherent thickness from the surface to the first pore, this wall being everywhere 
devoid of pores ; nor why, which comes to the same thing in the end, there does not exist 
a pore that is the last & nearest to the external surface ; this pore at least, if there were one, 
certainly has a wall that is free from pores & incapable of compression ; & here then the 
whole force of the argument used above applies perfectly unimpaired. 

22. Moreover, even if this idea is admitted, although it may be quite unintelligible, Violation of the 
then the whole force of the same argument applies to the first or last surface of the bodies ta^s'piace^any 
that are in immediate contact with one another ; or, if there are no continuous surfaces rate, in prime sur- 
congruent, then to the lines or points. For, whatever the manner may be in which contact 

takes place, there must be something in every case that certainly affords occasion for 
impenetrability, & causes the motion of the body that follows to be diminished, & that of 
the one in front to be increased. This, whatever it may be, from which the force of impene- 
trability is derived, at the instant at which immediate contact is obtained, must certainly 
change the velocity suddenly, & without any passage through intermediate stages ; & by 

4 8 


debet, atque labefactari, si ad ipsum immediatum contactum illo velocitatum discrimine 
deveniatur. Id vero est sane aliquid in quacunque e sententiis omnibus continuam 
extensionem tribuentibus materise. Est nimirum realis affectio qusedam corporis, videlicet 
ejus limes ultimus realis, superficies, realis superficiei limes linea, realis lineae limes punctum, 
qua affectiones utcunque in iis sententiis sint prorsus inseparabiles [n] ab ipso corpore, 
sunt tamen non utique intellectu confictae, sed reales, quas nimirum reales dimensiones 
aliquas habent, ut superficies binas, linea unam, ac realem motum, & translationem cum ipso 
corpore, cujus idcirco in iis sententiis debent, esse affectiones quaedam, vel modi. 

Objectio petita a 2 7. Est, qui dicat, nullum in iis committi saltum idcirco, quod censendum sit, nullum 

vucemassa, &,. J r . .. , ,, i\/r x 

motns. quae super- habere motum, superficiem, Imeam, punctum, quae massam habeant nullam. Motus, mquit, 

ficiebus, & punctis a Mechanicis habet pro mensura massam in velocitatem ductam : massa autem est super- 
non convemant. _.. , . i j /-^ 

ficies baseos ducta in crassitudmem, sive altitudmem, ex. gr. m pnsmatis. Quo minor est 
ejusmodi crassitude, eo minor est massa, & motus, ac ipsa crassitudine evanescente, evanescat 
oportet & massa, & motus. 

Kesponsionis ini- 24. Verum qui sic ratiocinatur, inprimis ludit in ipsis vocibus. Massam vulgo appellant 

tacam.^punctmn! quantitatem materiae, & motum corporum metiuntur per massam ejusmodi, ac velocitatem. 

posita extensione At quemadmodum in ipsa geometrica quantitate tria genera sunt quantitatum, corpus, vel 

contmua, e - ^11^^ q UO( J trinam dimensionem habet, superficies quae binas, linae, quae unicam, quibus 

accedit linese limes punctum, omni dimensione, & extensione carens ; sic etiam in Physica 

habetur in communi corpus tribus extensionis speciebus praeditum ; superficies realis extimus 

corporis limes, praedita binis ; linea, limes realis superficiei, habens unicam; & ejusdem 

lineae indivisibilis limes punctum. Utrobique alterum alterius est limes, non pars, & quatuor 

diversa genera constituunt. Superficies est nihil corporeum, sed non & nihil superficial, 

quin immo partes habet, & augeri potest, & minui ; & eodem pacto linea in ratione quidem 

superficiei est nihil, sed aliquid in ratione linese ; ac ipsum demum punctum est aliquid in 

suo genere, licet in ratione lineae sit nihil. 

QUO pacto nomen 25. Hinc autem in iis ipsis massa quaedam considerari potest duarum dimensionum, vel 

motus 'debeat 8 con- unius, vel etiam nullius continuae dimensionis, sed numeri punctorum tantummodo, uti 

venire superficie- quantitas ejus genere designetur ; quod si pro iis etiam usurpetur nomen massae generaliter, 

bus, imeis, punctis. motus q uan titas definiri poterit per productum ex velocitate, & massa ; si vero massae nomen 

tribuendum sit soli corpori, turn motus quidem corporis mensura erit massa in velocitatem 

ducta ; superficiei, lineae, punctorum quotcunque motus pro mensura habebit quantitatem 

superficiei, vel lineae, vel numerum punctorum in velocitatem ducta ; sed motus utique iis 

omnibus speciebus tribuendus erit, eruntque quatuor motuum genera, ut quatuor sunt 

quantitatum, solidi, superficiei, lineae, punctorum ; ac ut altera harum erit nihil in alterius 

ratione, non in sua ; ita alterius motus erit nihil in ratione alterius sed erit sane aliquid in 

ratione sui, non purum nihil. 

Fore, ut ea laedatur 
saltern in velocitate 

Motum passim r I2 i 2Q - g t q u id em jp S j Mechanici vulgo motum tribuunt & superficiebus & lineis, & 

tnbui punctis; ,'..* , . '. -m j 

fore, lit in eo ixda- punctis, ac centri gravitatis motum ubique nommant rhysici, quod centrum utique punctum 
i^ r Continuitatis est aliquod, non corpus trina praeditum dimensione, quam iste ad motus rationem, & 
appellationem requirit, ludendo, ut ajebam, in verbis. Porro in ejusmodi motibus exti- 
marum saltern superficierum, vel linearum, vel punctorum, saltus omnino committi debet, 
si ea ad contactum immediatum deveniant cum illo velocitatum discrimine, & continuitatis 
lex violari. 

27. Verum hac omni disquisitione omissa de notione motus, & massae, si factum ex 
velocitate, & massa, evanescente una e tribus dimensionibus, evanescit ; remanet utique 
velocitas reliquarum dimensionum, quae remanet, si eae reapse remanent, uti quidem omnino 
remanent in superficie, & ejus velocitatis mutatio haberi deberet per saltum, ac in ea violari 
continuitatis lex jam toties memorata. 

-, t i exin ?P<! n e- 28. Haec quidem ita evidentia sunt, ut omnino dubitari non possit, quin continuitatis 

trabilitate admissa ,.,..,/ .-KT i i j- j 

in minimis parti- lex infnngi debeat, & saltus m Naturam induci, ubi cum velocitatis discrimine ad se invicem 

cuiis. & ejus confu- acce dant corpora, & ad immediatum contactum deveniant, si modo impenetrabilitas 

corporibus tribuenda sit, uti revera est. Earn quidem non in integris tantummodo corpori- 

bus, sed in minimis etiam quibusque corporum particulis, atque elementis agnoverunt 

Physici universi. Fuit sane, qui post meam editam Theoriam, ut ipsam vim mei argument} 



that the Law of Continuity must be broken & destroyed, if immediate contact is arrived 
at with such a difference of velocity. Moreover, there is in truth always something of this 
sort in every one of the ideas that attribute continuous extension to matter. There is some 
real condition of the body, namely, its last real boundary, or its surface, a real boundary of 
a surface, a line, & a real boundary of a line, a point ; & these conditions, however insepar- 
able they may be in these theories from the body itself, are nevertheless certainly not 
fictions of the brain, but real things, having indeed certain real dimensions (for instance, a 
surface has two dimensions, & a line one) ; they also have real motion & movement of trans- 
lation along with the body itself ; hence in these theories they must be certain conditions 
or modes of it. 

23. Someone may say that there is no sudden change made, because it must be con- Objection derived 
sidered that a surface, a line or a point, having no mass, cannot have any motion. He may 1 mo/io^w^idi 
say that motion has, according to Mechanicians, as its measure, the mass multiplied by the do not accord with 
velocity ; also mass is the surface of the base multiplied by the thickness or the altitude, surfaces & P mis - 
as for instance in prisms. Hence the less the thickness, the less the mass & the motion ; 

thus, if the thickness vanishes, then both the mass & therefore the motion must vanish 
as well. 

24. Now the man who reasons in this manner is first of all merely playing with words. Commencement of 
Mass is commonly called quantity of matter, & the motion of bodies is measured by mass the answer to tl ? ls : 

. i * i . , * - . * , . c -_* cl SUrlclCC, OF ii 11116, 

of this kind & the velocity. But, just as in a geometrical quantity there are three kinds of or a point, is some- 
quantities, namely, a body or a solid having three dimensions, a surface with two, & a line \ 
with one : to which is added the boundary of a line, a point, lacking dimensions altogether, is supposed to ex- 
& of no extension. So also in Physics, a body is considered to be endowed with three lst ' 
species of extension ; a surface, the last real boundary of a body, to be endowed with two ; 
a line, the real boundary of a surface, with one ; & the indivisible boundary of the line, to 
be a point. In both subjects, the one is a boundary of the other, & not a part of it ; & 
they form four different kinds. There is nothing solid about a surface ; but that does not 
mean that there is also nothing superficial about it ; nay, it certainly has parts & can be 
increased or diminished. In the same way a line is nothing indeed when compared with 
a surface, but a definite something when compared with a line ; & lastly a point is a definite 
something in its own class, although nothing in comparison with a line. 

25. Hence also in these matters, a mass can be considered to be of two dimensions, or The manner in 
of one, or even of no continuous dimension, but only numbers of points, just as quantity of wn'^ma^and^the 
this kind is indicated. Now, if for these also, the term mass is employed in a generalized term motus is bound 
sense, we shall be able to define the quantity of motion by the product of the velocity & ! : ;^ p ^' to ? u j faces ' 

1 Ii ' 1 1 1 1 1 * 1 1 1 i HOPS, <X pOintS. 

the mass. But if the term mass is only to be used in connection with a solid body, then 
indeed the motion of a solid body will be measured by the mass multiplied by the velocity ; 
but the motion of a surface, or a line, or any number of points will have as their measure 
the quantity of the surface, or line, or the number- of the points, multiplied by the velocity. 
Motion at any rate will be ascribed in all these cases, & there will be four kinds of motion, 
as there are four kinds of quantity, namely, for a solid, a surface, a line, or for points ; and, as 
each class of the latter will be as nothing compared with the class before it, but something 
in its own class, so the motion of the one will be as nothing compared with the motion 
of the other, but yet really something, & not entirely nothing, compared with those of 
its own class. 

26. Indeed, Mechanicians themselves commonly ascribe motion to surfaces, lines & Motion is ascribed 
points, & Physicists universally speak of the motion of the centre of gravity ; this centre is minateiy 3 the'i^w 
undoubtedly some point, & not a body endowed with three dimensions, which the objector of Continuity is vio- 
demands for the idea & name of motion, by playing with words, as I said above. On the ' 

other hand, in this kind of motions of ultimate surfaces, or lines, or points, a sudden change 
must certainly be made, if they arrive at immediate contact with a difference of velocity 
as above, & the Law of Continuity must be violated. 

27. But, omitting all debate about the notions of motion & mass, if the product of it is at least a fact 
the velocity & the mass vanishes when one of the three dimensions vanish, there will still fated^tf^the^idea 
remain the velocity of the remaining dimensions ; & this will persist so long as the dimen- of the velocity of 
sions persist, as they do persist undoubtedly in the case of a surface. Hence the change P mts - 

in its velocity must have been made suddenly, & thereby the Law of Continuity, which I 
have already mentioned so many times, is violated. 

28. These things are so evident that it is absolutely impossible to doubt that the Law objection derived 

/./-! r - i i j j i . . j j . T iv from the admission 

of Continuity is infringed, & that a sudden change is introduced into Nature, when bodies O f impenetrability 
approach one another with a difference of velocity & come into immediate contact, if only in verv small P ar - 
we are to ascribe impenetrability to bodies, as we really should. And this property too, t ion. ' 
not in whole bodies only, but in any of the smallest particles of bodies, & in the elements as 
well, is recognized by Physicists universally. There was one, I must confess, who, after I 



infringeret, affirmavit, minimas corporum particulas post contactum superficierum com- 
penetrari non nihil, & post ipsam compenetrationem mutari velocitates per gradus. At id 
ipsum facile demonstrari potest contrarium illi inductioni, & analogiae, quam unam habemus 
in Physica investigandis generalibus naturae legibus idoneam, cujus inductionis vis quae sit, 
& quibus in locis usum habeat, quorum locorum unus est hie ipse impenetrabilitatis ad 
minimas quasque particulas extendendae, inferius exponam. 

Objectio a voce 

motus assumpta 

pro mutatione; 
confutatio ex 
reahtate motus 

2 Q. Fuit itidem e Leibnitianorum familia, qui post evulgatam Theoriam meam cen- 

. ' ,./>- , j- j j j -i 

suerit, dimcultatem ejusmodi amoveri posse dicendo, duas monades sibi etiam mvicem 

occurrentes cum velocitatibus quibuscunque oppositis aequalibus, post ipsum contactum 
..... . i, . . .' r ..... 

pergere moven sine locali progressione. Ham progressionem, ajebat, revera omnmo nihil 
esse, si a spatio percurso sestimetur, cum spatium sit nihil ; motum utique perseverare, & 
extingui per gradus, quia per gradus extinguatur energia ilia, qua in se mutuo agunt, sese 
premendo invicem. Is itidem ludit in voce motus, quam adhibet pro mutatione quacunque, 
& actione, vel actionis modo. Motus locaiis, & velocitas motus ipsius, sunt ea, quse ego 
quidem adhibeo, & quae ibi abrumpuntur per saltum. Ea, ut evidentissime constat, erant 
aliqua ante contactum, & post contactum mo-[i3]-mento temporis in eo casu abrumpuntur ; 
nee vero sunt nihil ; licet spatium pure imaginarium sit nihil. Sunt realis affectio rei 
mobilis fundata in ipsis modis localiter existendi, qui modi etiam relationes inducunt dis- 
tantiarum reales utique. Quod duo corpora magis a se ipsis invicem distent, vel minus ; 
quod localiter celerius moveantur, vel lentius ; est aliquid non imaginarie tantummodo, sed 
realiter diversum ; in eo vero per immediatum contactum saltus utique induceretur in eo 
casu, quo ego superius sum usus. 

Qui Continuitatu, 30. Et sane summus nostri aevi Geometra, & Philosophus Mac-Laurinus, cum etiam ipse 

jegem summover- co ni s j onem corporum contemplatus vidisset, nihil esse, quod continuitatis legem in collisione 
corporum facta per immediatum contactum conservare, ac tueri posset, ipsam continuitatis 
legem deferendam censuit, quam in eo casu omnino violari affirmavit in eo opere, quod de 
Newtoni Compertis inscripsit, lib. I, cap. 4. Et sane sunt alii nonnulli, qui ipsam con- 
tinuitatis legem nequaquam admiserint, quos inter Maupertuisius, vir celeberrimus, ac de 
Republica Litteraria optime meritus, absurdam etiam censuit, & quodammodo inexplica- 
bilem. Eodem nimirum in nostris de corporum collisione contemplationibus devenimus 
Mac-Laurinus, & ego, ut viderimus in ipsa immediatum contactum, atque impulsionem cum 
continuitatis lege conciliari non posse. At quoniam de impulsione, & immediate corporum 
contactu ille ne dubitari quidem posse arbitrabatur, (nee vero scio, an alius quisquam omnem 
omnium corporum immediatum contactum subducere sit ausus antea, utcunque aliqui aeris 
velum, corporis nimirum alterius, in collisione intermedium retinuerint) continuitatis 
legem deseruit, atque infregit. 

Theorise exortus, 
t^'t U f fien 

31. Ast ego cum ipsam continuitatis legem aliquanto diligentius considerarim, & 
, quibus ea innititur, perpenderim, arbitratus sum, ipsam omnino e Natura 
submoveri non posse, qua proinde retenta contactum ipsum immediatum submovendum 
censui in collisionibus corporum, ac ea consectaria persecutus, quae ex ipsa continuitate 
servata sponte profluebant, directa ratiocinatione delatus sum ad earn, quam superius 
exposui, virium mutuarum legem, quae consectaria suo quaeque ordine proferam, ubi ipsa, 
quae ad continuitatis legem retinendam argumenta me movent, attigero. 

Lex Continuitatis 32. Continuitatis lex, de qua hie agimus, in eo sita est, uti superius innui, ut quaevis 

quid sit : discn- j i i- T i- r 

men inter status, quantitas, dum ab una magmtudme ad aliam migrat, debeat transire per omnes intermedias 
& incrementa. ejusdem generis magnitudines. Solet etiam idem exprimi nominandi transitum per gradus 
intermedios, quos quidem gradus Maupertuisius ita accepit, quasi vero quaedam exiguae 
accessiones fierent momento temporis, in quo quidem is censuit violari jam necessario legem 
ipsam, quae utcunque exiguo saltu utique violatur nihilo minus, quam maximo ; cum 
nimi-[l4]-rum magnum, & parvum sint tantummodo respectiva ; & jure quidem id censuit ; 
si nomine graduum incrementa magnitudinis cujuscunque momentanea intelligerentur. 


had published my Theory, endeavoured to overcome the force of the argument I had used 
by asserting that the minute particles of the bodies after contact of the surfaces were 
subject to compenetration in some measure, & that after compenetration the velocities 
were changed gradually. But it can be easily proved that this is contrary to that induction 
& analogy, such as we have in Physics, one peculiarly adapted for the investigation of the 
general laws of Nature. What the power of this induction is, & where it can be used (one 
of the cases is this very matter of extending impenetrability to the minute particles of a 
body), I will set forth later. 

29. There was also one of the followers of Leibniz who, after I had published my Objection to the 
Theory, expressed his opinion that this kind of difficulty could be removed by saying that used fora"change^ 
two monads colliding with one another with any velocities that were equal & opposite refutation from the 

,,,., .. . .-I , , TT reality of local mo- 

would, alter they came into contact, go on moving without any local progression, rle tion. 
added that that progression would indeed be absolutely nothing, if it were estimated by the 
space passed over, since the space was nothing ; but the motion would go on & be destroyed 
by degrees, because the energy with which they act upon one another, by mutual pressure, 
would be gradually destroyed. He also is playing with the meaning of the term motus, 
which he uses both for any change, & for action & mode of action. Local motion, & the 
velocity of that motion are what I am dealing with, & these are here broken off suddenly. 
These, it is perfectly evident, were something definite before contact, & after contact in 
an instant of time in this case they are broken off. Not that they are nothing ; although 
purely imaginary space is nothing. They are real conditions of the movable thing 
depending on its modes of extension as regards position ; & these modes induce relations 
between the distances that are certainly real. To account for the fact that two bodies 
stand at a greater distance from one another, or at a less ; or for the fact that they are 
moved in position more quickly, or more slowly ; to account for this there must be some- 
thing that is not altogether imaginary, but real & diverse. In this something there would 
be induced, in the question under consideration, a sudden change through immediate 

30. Indeed the finest geometrician & philosopher of our times, Maclaurin, after he too There are some who 
had considered the collision of solid bodies & observed that there is nothing which could i^ d oi continuity 5 
maintain & preserve the Law of Continuity in the collision of bodies accomplished by 

immediate contact, thought that the Law of Continuity ought to be abandoned. He 
asserted that, in general in the case of collision, the law was violated, publishing his idea in 
the work that he wrote on the discoveries of Newton, bk. i, chap. 4. True, there are some 
others too, who would not admit the Law of Continuity at all ; & amongst these, Mauper- 
tuis, a man of great reputation & the highest merit in the world of letters, thought it was 
senseless, & in a measure inexplicable. Thus, Maclaurin came to the same conclusion as 
myself with regard to our investigations on the collision of bodies ; for we both saw that, in 
collision, immediate contact & impulsive action could not be reconciled with the Law of 
Continuity. But, whereas he came to the conclusion that there could be no doubt about 
the fact of impulsive action & immediate contact between the bodies, he impeached & 
abrogated the Law of Continuity. Nor indeed do I know of anyone else before me, who 
has had the courage to deny the existence of all immediate contact for any bodies whatever, 
although there are some who would retain a thin layer of air, (that is to say, of another body), 
in between the two in collision. 

31. But I, after considering the Law of Continuity somewhat more carefully, & Th e origin of my 
pondering over the fundamental ideas on which it depends, came to the conclusion that thisLaw, as'shouid 
it certainly could not be withdrawn altogether out of Nature. Hence, since it had to be be done, 
retained, I came to the conclusion that immediate contact in the collision of solid bodies 

must be got rid of ; &, investigating the deductions that naturally sprang from the 
conservation of continuity, I was led by straightforward reasoning to the law that I have set 
forth above, namely, the law of mutual forces. These deductions, each set out in order, 
I will bring forward when I come to touch upon those arguments that persuade me to 
retain the Law of Continuity. 

32. The Law of Continuity, as we here deal with it, consists in the idea that, as I J he nature of the 

. j , ..''... . , , Law of Continuity ; 

intimated above, any quantity, in passing from one magnitude to another, must pass through distinction between 
all intermediate magnitudes of the same class. The same notion is also commonly expressed stat< ~ s & incre - 

, , .,,. ,. -11 ments. 

by saying that the passage is made by intermediate stages or steps ; these steps indeed 
Maupertuis accepted, but considered that they were very small additions made in an 
instant of time. In this he thought that the Law of Continuity was already of necessity 
violated, the law being indeed violated by any sudden change, no matter how small, in no 
less a degree than by a very great one. For, of a truth, large & small are only relative terms ; 
& he rightly thought as he did, if by the name of steps we are to understand momentaneous 


Geometriae usus ad 
earn exponendam : 
momenta punctis, 
tempera continua 
lineis expressa. 

Fluxus ordinatae 
transeuntis per 
m agnit u d i nes 
omnes intermedias. 

Idem in quantitate 
variabili expressa : 
aequivocatio in 
voce gradus. 

FKMH K' M' D' 

FIG. 3. 

Verum id ita intelligendum est ; ut singulis momentis singuli status respondeant ; incre- 
menta, vel decrementa non nisi continuis tempusculis. 

33. Id sane admodum facile concipitur ope Geometriae. Sit recta quaedam AB in 
fig. 3, ad quam referatur quaedam alia linea CDE. Exprimat prior ex iis tempus, uti solet 
utique in ipsis horologiis circularis peripheria 

ab indicis cuspide denotata tempus definire. 
Quemadmodum in Geometria in lineis 
puncta sunt indivisibiles limites continuarum 
lineas partium, non vero partes linese ipsius ; 
ita in tempore distinguenda; erunt partes 
continui temporis respondentes ipsis lines 
partibus, continue itidem & ipsas, a mo- 
mentis, quae sunt indivisibiles earum partium 
limites, & punctis respondent ; nee inpos- 
terum alio sensu agens de tempore momenti 
nomen adhibebo, quam eo indivisibilis 
limitis ; particulam vero temporis utcunque 
exiguam, & habitam etiam pro infinitesima, 
tempusculum appellabo. 

34. Si jam a quovis puncto rectae AB, ut F, H, erigatur ordinata perpendicularis FG, 
HI, usque ad lineam CD ; ea poterit repraesentare quantitatem quampiam continuo 
variabilem. Cuicunque momento temporis F, H, respondebit sua ejus quantitatis magnitudo 
FG, HI ; momentis autem intermediis aliis K, M, aliae magnitudines, KL, MN, respondebunt ; 
ac si a puncto G ad I continua, & finita abeat pars linese CDE, facile patet & accurate de- 
monstrari potest, utcunque eadem contorqueatur, nullum fore punctum K intermedium, 
cui aliqua ordinata KL non respondeat ; & e converse nullam fore ordinatam magnitu- 
dinis intermediae inter FG, HI, quae alicui puncto inter F, H intermedio non respondeat. 

35. Quantitas ilia variabilis per hanc variabilem ordinatam expressa mutatur juxta 
continuitatis legem, quia a magnitudine FG, quam habet momento temporis F, ad magni- 
tudinem HI, quae respondet momento temporis H, transit per omnes intermedias magnitu- 
dines KL, MN, respondentes intermediis momentis K, M, & momento cuivis respondet 
determinata magnitudo. Quod si assumatur tempusculum quoddam continuum KM 
utcunque exiguum ita, ut inter puncta L, N arcus ipse LN non mutet recessum a recta AB 
in accessum ; ducta LO ipsi parallela, habebitur quantitas NO, quas in schemate exhibito 
est incrementum magnitudinis ejus quantitatis continuo variatae. Quo minor est ibi 
temporis particula KM, eo minus est id incrementum NO, & ilia evanescente, ubi congruant 
momenta K, M, hoc etiam evanescit. Potest quaevis magnitudo KL, MN appellari status 
quidam variabilis illius quantitatis, & gradus nomine deberet potius in-[i5]-telligi illud 
incrementum NO, quanquam aliquando etiam ille status, ilia magnitudo KL nomine gradus 
intelligi solet, ubi illud dicitur, quod ab una magnitudine ad aliam per omnes intermedios 
gradus transeatur ; quod quidem aequivocationibus omnibus occasionem exhibuit. 

status singuios 36. Sed omissis aequivocationibus ipsis, illud, quod ad rem facit, est accessio incremen- 

menta^vero'utcun" torum facta non momento temporis, sed tempusculo continuo, quod est particula continui 

que parva tem- temporis. Utcunque exiguum sit incrementum ON, ipsi semper respondet tempusculum 

respondere C ntinuis q. u ddam KM continuum. Nullum est in linea punctum M ita proximum puncto K, ut sit 

primum post ipsum ; sed vel congruunt, vel intercipiunt lineolam continua bisectione per 

alia intermedia puncta perpetuo divisibilem in infinitum. Eodem pacto nullum est in 

tempore momentum ita proximum alteri praecedenti momento, ut sit primum post ipsum, 

sed vel idem momentum sunt, vel inter jacet inter ipsa tempusculum continuum per alia 

intermedia momenta divisibile in infinitum ; ac nullus itidem est quantitatis continuo 

variabilis status ita proximus praecedenti statui, ut sit primus post ipsum accessu aliquo 

momentaneo facto : sed differentia, quae inter ejusmodi status est, debetur intermedio 

continuo tempusculo ; ac data lege variationis, sive natura lineae ipsam exprimentis, & 

quacunque utcunque exigua accessione, inveniri potest tempusculum continuum, quo ea 

accessio advenerit. 

Transitus sine sal- 37- Atque sic quidem intelligitur, quo pacto fieri possit transitus per intermedias 

tu, etiamapositivis magnitudines omnes, per intermedios status, per gradus intermedios, quin ullus habeatur 

ad negativa perm- , . r -, . ... , ' " 

hiium, quod tamen saltus utcunque exiguus momento temporis factus. Notari mud potest tantummodo, 
m" eS s t ed Vere u 'ida 1 m mutat i nem ner i alicubi per incrementa, ut ubi KL abit, in MN per NO ; alicubi per 
reaiis status, 1 " ' decrementa, ut ubi K'L' abeat in N'M' per O'N' ; quin immo si linea CDE, quse legem 


increments of any magnitude whatever. But the idea should be interpreted as follows : 
single states correspond to single instants of time, but increments or decrements only to 
small intervals of continuous time. 

33. The idea can be very easily assimilated by the help of geometry. Explanation by the 
Let AB be any straight line (Fig. 3), to which as axis let any other line CDE be referred. " n s s e ta t f s ^eTes^ 

Let the first of them represent the time, in the same manner as it is customary to specify ted by points, con- 
the time in the case of circular clocks by marking off the periphery with the end of a pointer. 1^ "^s*** f 
Now, just as in geometry, points are the indivisible boundaries of the continuous parts of 
a line, so, in time, distinction must be made between parts of continuous time, which cor- 
respond to these parts of a line, themselves also continuous, & instants of time, which are 
the indivisible boundaries of those parts of time, & correspond to points. In future I shall 
not use the term instant in any other sense, when dealing with time, than that of the 
indivisible boundary ; & a small part of time, no matter how small, even though it is 
considered to be infinitesimal, I shall term a tempuscule, or small interval of time. 

34. If now from any points F,H on the straight line AB there are erected at right angles T . he flux . f the or ~ 
to it ordinates FG, HI, to meet the line CD ; any of these ordinates can be taken to repre- through^ ail *inter S 
sent a quantity that is continuously varying. To any instant of time F, or H, there will mediate values, 
correspond its own magnitude of the quantity FG, or HI ; & to other intermediate instants 

K, M, other magnitudes KL, MN will correspond. Now, if from the point G, there pro- 
ceeds a continuous & finite part of the line CDE, it is very evident, & it can be rigorously 
proved, that, no matter how the curve twists & turns, there is no intermediate point K, 
to which some ordinate KL does not correspond ; &, conversely, there is no ordinate of 
magnitude intermediate between FG & HI, to which there does not correspond a point 
intermediate between F & H. 

35. The variable quantity that is represented by this variable ordinate is altered in The same holds 
accordance with the Law of Continuity ; for, from the magnitude FG, which it has at able 1 quantity w 
the instant of time F, to the magnitude HI, which corresponds to the instant H, it passes represented ; equi- 
through all intermediate magnitudes KL, MN, which correspond to the intermediate oUhe 1( term Itep^ 
instants K, M ; & to every instant there corresponds a definite magnitude. But if we take 

a definite small interval of continuous time KM, no matter how small, so that between the 
points L & N the arc LN does not alter from recession from the line AB to approach, & 
draw LO parallel to AB, we shall obtain the quantity NO that in the figure as drawn is the 
increment of the magnitude of the continuously varying quantity. Now the smaller the 
interval of time KM, the smaller is this increment NO ; & as that vanishes when the 
instants of time K, M coincide, the increment NO also vanishes. Any magnitude KL, MN 
can be called a state of the variable quantity, & by the name step we ought rather to under- 
stand the increment NO ; although sometimes also the state, or the magnitude KL is 
accustomed to be called by the name step. For instance, when it is said that from one 
magnitude to another there is a passage through all intermediate stages or steps ; but this 
indeed affords opportunity for equivocations of all sorts. 

36. But, omitting all equivocation of this kind, the point is this : that addition of single states cor- 

.' 1-11 < i 11 . respond to instants, 

increments is accomplished, not m an instant 01 time, but in a small interval of con- but increments 
tinuous time, which is a part of continuous time. However small the increment ON may however sma11 to 

i i i Tru if mi > intervals of con- 

DC, there always corresponds to it some continuous interval KM. 1 here is no point M tinuous time. 

in the straight line AB so very close to the point K, that it is the next after it ; but either 
the points coincide, or they intercept between them a short length of line that is divisible 
again & again indefinitely by repeated bisection at other points that are in between M & 
K. In the same way, there is no instant of time that is so near to another instant that has 
gone before it, that it is the next after it ; but either they are the same instant, or there 
lies between them a continuous interval that can be divided indefinitely at other inter- 
mediate instants. Similarly, there is no state of a continuously varying quantity so very 
near to a preceding state that it is the next state to it, some momentary addition having 
been made ; any difference that exists between two states of the same kind is due to a 
continuous interval of time that has passed in the meanwhile. Hence, being given the 
law of variation, or the nature of the line that represents it, & any increment, no matter 
how small, it is possible to find a small interval of continuous time in which the increment 
took place. 

37. In this manner we can understand how it is possible for a passage to take place Passages without 
through all intermediate magnitudes, through intermediate states, or through intermediate from^positive 1 8 to 
stages, without any sudden change being made, no matter how small, in an instant of time, negative through 

T' 11 1111 i 111- /i zero : zero how- 

It can merely be remarked that change in some places takes place by increments (as when ever ; s not real i y 

KL becomes MN by the addition of NO), in other places by decrements (as when K'L' nothing, but acer- 

' tain real state. 


variationis exhibit, alicubi secet rectam, temporis AB, potest ibidem evanescere magnitude, 
ut ordinata M'N', puncto M' allapso ad D evanesceret, & deinde mutari in negativam PQ, 
RS, habentem videlicet directionem contrariam, quae, quo magis ex oppositae parte crescit, 
eo minor censetur in ratione priore, quemadmodum in ratione possessionis, vel divitiarum, 
pergit perpetuo se habere pejus, qui iis omnibus, quae habebat, absumptis, aes alienum 
contrahit perpetuo majus. Et in Geometria quidem habetur a positivo ad negativa 
transitus, uti etiam in Algebraicis formulis, tarn transeundo per nihilum, quam per innnitum, 
quos ego transitus persecutus sum partim in dissertatione adjecta meis Sectionibus Conicis, 
partim in Algebra 14, & utrumque simul in dissertatione De Lege Continuitatis ; sed in 
Physica, ubi nulla quantitas in innnitum excrescit, is casus locum non habet, & non, nisi 
transeundo per nihilum, transitus fit a positi-[i6]-vis ad negativa, ac vice versa ; quanquam, 
uti inferius innuam, id ipsum sit non nihilum revera in se ipso, sed realis quidem status, & 
habeatur pro nihilo in consideration quadam tantummodo, in qua negativa etiam, qui sunt 
veri status, in se positivi, ut ut ad priorem seriem pertinentes negative quodam modo, 
negativa appellentur. 

Proponitur pro- ,_ Exposita hoc pacto, & vindicata continuitatis lege, earn in Natura existere plerique 

banda existentia _, ., J . . r . . .... ... P . . ,-, r . 

legis Continuitat.s. Philosophi arbitrantur, contradicentibus nonnullis, uti supra mnui. Ego, cum in earn 
primo inquirerem, censui, eandem omitti omnino non posse ; si earn, quam habemus unicam, 
Naturae analogiam, & inductionis vim consulamus, ope cujus inductionis earn demonstrare 
conatus sum in pluribus e memoratis dissertationibus, ac eandem probationem adhibet 
Benvenutus in sua Synopsi Num. 119; in quibus etiam locis, prout diversis occasionibus 
conscripta sunt, repetuntur non nulla. 

Ejus probatio ab ,g Longum hie esset singula inde excerpere in ordinem redacta : satis erit exscribere 

mductione satis ,. Jy . _ , ~ . P . r ,-, -n i j 

ampia. dissertatioms De lege Continuitatis numerum 138. Post mductionem petitam praecedente 

numero a Geometria, quae nullum uspiam habet saltum, atque a motu locali, in quo nunquam 
ab uno loco ad alium devenitur, nisi ductu continue aliquo, unde consequitur illud, dis- 
tantiam a dato loco nunquam mutari in aliam, neque densitatem, quae utique a distantiis 
pendet particularum in aliam, nisi transeundo per intermedias ; fit gradus in eo numero ad 
motuum velocitates, & ductus, quas magis hie ad rem faciunt, nimirum ubi de velocitate 
agimus non mutanda per saltum in corporum collisionibus. Sic autem habetur : " Quin 
immo in motibus ipsis continuitas servatur etiam in eo, quod motus omnes in lineis continuis 
fiunt nusquam abruptis. Plurimos ejusmodi motus videmus. Planetae, & cometse in lineis 
continuis cursum peragunt suum, & omnes retrogradationes fiunt paullatim, ac in stationibus 
semper exiguus quidem motus, sed tamen habetur semper, atque hinc etiam dies paullatim 
per auroram venit, per vespertinum crepusculum abit, Solis diameter non per saltum, sed 
continuo motu supra horizontem ascendit, vel descendit. Gravia itidem oblique projecta 
in lineis itidem pariter continuis motus exercent suos, nimirum in parabolis, seclusa ^aeris 
resistentia, vel, ea considerata, in orbibus ad hyperbolas potius accedentibus, & quidem 
semper cum aliqua exigua obliquitate projiciuntur, cum infinities infinitam improbabilitatem 
habeat motus accurate verticalis inter infinities infinitas inclinationes, licet exiguas, & sub 
sensum non cadentes, fortuito obvenienfe, qui quidem motus in hypothesi Telluris^motae a 
parabolicis plurimum distant, & curvam continuam exhibent etiam pro casu projectionis 
accurate verticalis, quo, quiescente penitus Tellure, & nulla ventorum vi deflectente motum, 
haberetur [17] ascensus rectilineus, vel descensus. Immo omnes alii motus a gravitate 
pendentes, omnes ab elasticitate, a vi magnetica, continuitatem itidem servant ; cum earn 
servent vires illse ipsae, quibus gignuntur. Nam gravitas, cum decrescat in ratione reciproca 
duplicata distantiarum, & distantise per saltum mutari non possint, mutatur per omnes 
intermedias magnitudines. Videmus pariter, vim magneticam a distantiis pendere lege 
continua ; vim elasticam ab inflexione, uti in laminis, vel a distantia, ut in particulis aeris 
compressi. In iis, & omnibus ejusmodi viribus, & motibus, quos gignunt, continuitas habetur 
semper, tarn in lineis quae describuntur, quam in velocitatibus, quae pariter per omnes 
intermedias magnitudines mutantur, ut videre est in pendulis, in ascensu corporum gravium, 


becomes N'M' by the subtraction of O'N') ; moreover, if the line CDE, which represents 
the law of variation, cuts the straight AB, which is the axis of time, in any point, then the 
magnitude can vanish at that point (just as the ordinate M'N' would vanish when the 
point M' coincided with D), & be changed into a negative magnitude PQ, or RS, that is 
to say one having an opposite direction ; & this, the more it increases in the opposite sense, 
the less it is to be considered in the former sense (just as in the idea of property or riches, 
a man goes on continuously getting worse off, when, after everything he had has been 
taken away from him, he continues to get deeper & deeper into debt). In Geometry too 
we have this passage from positive to negative, & also in algebraical formulae, the passage 
being made not only through nothing, but also through infinity ; such I have discussed, 
the one in a dissertation added to my Conic Sections, the other in my Algebra ( 14), & both 
of them together in my essay De Lege Continuitatis ; but in Physics, where no quantity 
ever increases to an infinite extent, the second case has no place ; hence, unless the passage 
is made through the value nothing, there is no passage from positive to negative, or vice 
versa. Although, as I point out below, this nothing is not really nothing in itself, but a 
certain real state ; & it may be considered as nothing only in a certain sense. In the same 
sense, too, negatives, which are true states, are positive in themselves, although, as they 
belong to the first set in a certain negative way, they are called negative. 

38. Thus explained & defended, the Law of Continuity is considered by most philoso- I propose to prove 
phers to exist in Nature, though there are some who deny it, as I mentioned above. I, LaVof^Continuity 6 
when first I investigated the matter, considered that it was absolutely impossible that it 
should be left out of account, if we have regard to the unparalleled analogy that there is 
with Nature & to the power of induction ; & by the help of this induction I endeavoured 
to prove the law in several of the dissertations that I have mentioned, & Benvenutus also 
used the same form of proof in his Synopsis (Art. 119). In these too, as they were written 
on several different occasions, there are some repetitions. 

39. It would take too long to extract & arrange in order here each of the passages in Proof by induction 
these essays ; it will be sufficient if I give Art. 138 of the dissertation De Lege Continuitatis. s ~^^ for the 
After induction derived in the preceding article from geometry, in which there is no sudden 
change anywhere, & from local motion, in which passage from one position to another 
never takes place unless by some continuous progress (the consequence of which is that a 
distance from any given position can never be changed into another distance, nor the 
density, which depends altogether on the distances between the particles, into another density, 
except by passing through intermediate stages), the step is made in that article to the 
velocities of motions, & deductions, which have more to do with the matter now in hand, 
namely, where we are dealing with the idea that the velocity is not changed suddenly in the 
collision of solid bodies. These are the words : " Moreover in motions themselves 
continuity is preserved also in the fact that all motions take place in continuous lines that 
are not broken anywhere. We see a great number of motions of this kind. The planets & 
the comets pursue their courses, each in its own continuous line, & all retrogradations are 
gradual ; & in stationary positions the motion is always slight indeed, but yet there is 
always some ; hence also daylight comes gradually through the dawn, & goes through the 
evening twilight, as the diameter of the sun ascends above the horizon, not suddenly, but 
by a continuous motion, & in the same manner descends. Again heavy bodies projected 
obliquely follow their courses in lines also that are just as continuous ; namely, in para- 
bolae, if we neglect the resistance of the air, but if that is taken into account, then in orbits 
that are more nearly hyperbolae. Now, they are always projected with some slight obli- 
quity, since there is an infinitely infinite probability against accurate vertical motion, from 
out of the infinitely infinite number of inclinations (although slight & not capable of being 
observed), happening fortuitously. These motions are indeed very far from being para- 
bolae, if the hypothesis that the Earth is in motion is adopted. They give a continuous 
curve also for the case of accurate vertical projection, in which, if the Earth were at rest, 
& no wind-force deflected the motion, rectilinear ascent & descent would be obtained. 
All other motions that depend on gravity, all that depend upon elasticity, or magnetic 
force, also preserve continuity ; for the forces themselves, from which the motions arise, 
preserve it. For gravity, since it diminishes in the inverse ratio of the squares of the dis- 
tances, & the distances cannot be changed suddenly, is itself changed through every inter- 
mediate stage. Similarly we see that magnetic force depends on the distances according 
to a continuous law ; that elastic force depends on the amount of bending as in plates, or 
according to distance as in particles of compressed air. In these, & all other forces of the 
sort, & in the motions that arise from them, we always get continuity, both as regards the 
lines which they describe & also in the velocities which are changed in similar manner 
through all intermediate magnitudes ; as is seen in pendulums, in the ascent of heavy 


& in aliis mille ejusmodi, in quibus mutationes velocitatis fiunt gradatim, nee retro cursus 
reflectitur, nisi imminuta velocitate per omnes gradus. Ea diligentissime continuitatem 
servat omnia. Hinc nee ulli in naturalibus motibus habentur anguli, sed semper mutatio 
directionis fit paullatim, nee vero anguli exacti habentur in corporibus ipsis, in quibus 
utcunque videatur tennis acies, vel cuspis, microscopii saltern ope videri solet curvatura, 
quam etiam habent alvei fluviorum semper, habent arborum folia, & frondes, ac rami, habent 
lapides quicunque, nisi forte alicubi cuspides continuae occurrant, vel primi generis, quas 
Natura videtur affectare in spinis, vel secundi generis, quas videtur affectare in avium 
unguibus, & rostro, in quibus tamen manente in ipsa cuspide unica tangente continuitatem 
servari videbimus infra. Infinitum esset singula persequi, in quibus continuitas in Natura 
observatur. Satius est generaliter provocare ad exhibendum casum in Natura, in quo 
eontinuitas non servetur, qui omnino exhiberi non poterit." 

Duplex inductionis 40. Inductio amplissima turn ex hisce motibus, ac velocitatibus, turn ex aliis pluribus 

vimhabeatittductio exem P n ' s > <l U3e habemus in Natura, in quibus ea ubique, quantum observando licet depre- 
incompieta. hendere, continuitatem vel observat accurate, vel affcctat, debet omnino id efficere, ut ab 

ea ne in ipsa quidem corporum collisione recedamus. Sed de inductionis natura, & vi, ac 
ejusdem usu in Physica, libet itidem hie inserere partem numeri 134, & totum 135, disserta- 
tionis De Lege Continuitatis. Sic autem habent ibidem : " Inprimis ubi generales Naturae 
leges investigantur, inductio vim habet maximam, & ad earum inventionem vix alia ulla 
superest via. Ejus ope extensionem, figurabilitem, mobilitatem, impenetrabilitatem 
corporibus omnibus tribuerunt semper Philosophi etiam veteres, quibus eodem argumento 
inertiam, & generalem gravitatem plerique e recentioribus addunt. Inductio, ut demon- 
strationis vim habeat, debet omnes singulares casus, quicunque haberi possunt percurrere. 
Ea in Natu-[i8]-rae legibus stabiliendis locum habere non potest. Habet locum laxior 
qusedam inductio, quae, ut adhiberi possit, debet esse ejusmodi, ut inprimis in omnibus iis 
casibus, qui ad trutinam ita revocari possunt, ut deprehendi debeat, an ea lex observetur, 
eadem in iis omnibus inveniatur, & ii non exiguo numero sint ; in reliquis vero, si quse prima 
fronte contraria videantur, re accuratius perspecta, cum ilia lege possint omnia conciliari ; 
licet, an eo potissimum pacto concilientur, immediate innotescere, nequaquam possit. Si 
eae conditiones habeantur ; inductio ad legem stabiliendam censeri debet idonea. Sic quia 
videmus corpora tarn multa, quae habemus prae manibus, aliis corporibus resistere, ne in 
eorum locum adveniant, & loco cedere, si resistendo sint imparia, potius, quam eodem 
perstare simul ; impenetrabilitatem corporum admittimus ; nee obest, quod qusedam 
corpora videamus intra alia, licet durissima, insinuari, ut oleum in marmora, lumen in 
crystalla, & gemmas. Videmus enim hoc phsenomenum facile conciliari cum ipsa impene- 
trabilitate, dicendo, per vacuos corporum poros ea corpora permeare. (Num. 135). 
Praeterea, qusecunque proprietates absolutae, nimirum quae relationem non habent ad 
nostros sensus, deteguntur generaliter in massis sensibilibus corporum, easdem ad quascunque 
utcunque exiguas particulas debemus transferre ; nisi positiva aliqua ratio obstet, & nisi sint 
ejusmodi, quae pendeant a ratione totius, seu multitudinis, contradistincta a ratione partis. 
Primum evincitur ex eo, quod magna, & parva sunt respectiva, ac insensibilia dicuntur ea, 
quse respectu nostrae molis, & nostrorum sensuum sunt exigua. Quare ubi agitur de 
proprietatibus absolutis non respectivis, quaecunque communia videmus in iis, quse intra 
limites continentur nobis sensibiles, ea debemus censere communia etiam infra eos limites : 
nam ii limites respectu rerum, ut sunt in se, sunt accidentales, adeoque siqua fuisset analogise 
Isesio, poterat ilia multo facilius cadere intra limites nobis sensibiles, qui tanto laxiores sunt, 
quam infra eos, adeo nimirum propinquos nihilo. Quod nulla ceciderit, indicio est, nullam 
esse. Id indicium non est evidens, sed ad investigationis principia pertinet, quae si juxta 


bodies, & in a thousand other things of the same kind, where the changes of velocity occur 
gradually, & the path is not retraced before the velocity has been diminished through all 
degrees. All these things most strictly preserve continuity. Hence it follows that no 
sharp angles are met with in natural motions, but in every case a change of direction occurs 
gradually ; neither do perfect angles occur in bodies themselves, for, however fine an edge 
or point in them may seem, one can usually detect curvature by the help of the microscope 
if nothing else. We have this gradual change of direction also in the beds of rivers, in the 
leaves, boughs & branches of trees, & stones of all kinds ; unless, in some cases perchance, 
there may be continuous pointed ends, either of the first kind, which Nature is seen to 
affect in thorns, or of the second kind, which she is seen to do in the claws & the beak of 
birds ; in these, however, we shall see below that continuity is still preserved, since we are 
left with a single tangent at the extreme end. It would take far too long to mention every 
single thing in which Nature preserves the Law of Continuity ; it is more than sufficient 
to make a general statement challenging the production of a single case in Nature, in which 
continuity is not preserved ; for it is absolutely impossible for any such case to be brought 

40. The effect of the very complete induction from such motions as these & velocities, induction of a two- 
as well as from a large number of other examples, such as we have in Nature, where Nature * old , kil \ d ' when 

e c ,...-..& why incomplete 

in every case, as far as can be gathered from direct observation, maintains continuity or induction has vaii- 

tries to do so, should certainly be that of keeping us from neglecting it even in the case 

of collision of bodies. As regards the nature & validity of induction, & its use in Physics, 

I may here quote part of Art. 134 & the wjiole of Art. 135 from my dissertation De Lege 

Continuitatis, The passage runs thus : " Especially when we investigate the general laws 

of Nature, induction has very great power ; & there is scarcely any other method beside 

it for the discovery of these laws. By its assistance, even the ancient philosophers attributed 

to all bodies extension, figurability, mobility, & impenetrability ; & to these properties, 

by the use of the same method of reasoning, most of the later philosophers add inertia & 

universal gravitation. Now, induction should take account of every single case that can 

possibly happen, before it can have the force of demonstration ; such induction as this has no 

place in establishing the laws of Nature. But use is made of an induction of a less rigorous 

type ; in order that this kind of induction may be employed, it must be of such a nature 

that in all those cases particularly, which can be examined in a manner that is bound to 

lead to a definite conclusion as to whether or no the law in question is followed, in all of 

them the same result is arrived at ; & that these cases are not merely a few. Moreover, 

in the other cases, if those which at first sight appeared to be contradictory, on further & 

more accurate investigation, can all of them be made to agree with the law ; although, 

whether they can be made to agree in this way better than in any other whatever, it is 

impossible to know directly anyhow. If such conditions obtain, then it must be considered 

that the induction is adapted to establishing the law. Thus, as we see that so many of 

the bodies around us try to prevent other bodies from occupying the position which they 

themselves occupy, or give way to them if they are not capable of resisting them, rather 

than that both should occupy the same place at the same time, therefore we admit the 

impenetrability of bodies. Nor is there anything against the idea in the fact that we see 

certain bodies penetrating into the innermost parts of others, although the latter are very 

hard bodies ; such as oil into marble, & light into crystals & gems. For we see that this 

phenomenon can very easily be reconciled with the idea of impenetrability, by supposing 

that the former bodies enter and pass through empty pores in the latter bodies (Art. 

135). In addition, whatever absolute properties, for instance those that bear no relation 

to our senses, are generally found to exist in sensible masses of bodies, we are bound to 

attribute these same properties also to all small parts whatsoever, no matter how small 

they may be. That is to say, unless some positive reason prevents this ; such as that they 

are of such a nature that they depend on argument having to do with a body as a whole, 

or with a group of particles, in contradistinction to an argument dealing with a part only. 

The proof comes in the first place from the fact that great & small are relative terms, & 

those things are called insensible which are very small with respect to our own size & with 

regard to our senses. Therefore, when we consider absolute, & not relative, properties, 

whatever we perceive to be common to those contained within the limits that are sensible 

to us, we should consider these things to be still common to those beyond those limits. 

For these limits, with regard to such matters as are self-contained, are accidental ; & thus, 

if there should be any violation of the analogy, this would be far more likely to happen 

between the limits sensible to us, which are more open, than beyond them, where indeed 

they are so nearly nothing. Because then none did happen thus, it is a sign that there is 

none. This sign is not evident, but belongs to the principles of investigation, which 

generally proves successful if it is carried out in accordance with certain definite wisely 


quasdam prudentes regulas fiat, successum habere solet. Cum id indicium fallere possit ; 
fieri potest, ut committatur error, sed contra ipsum errorem habebitur praesumptio, ut 
etiam in jure appellant, donee positiva ratione evincatur oppositum. Hinc addendum fuit, 
nisi ratio positiva obstet. Sic contra hasce regulas peccaret, qui diceret, corpora quidem 
magna compenetrari, ac replicari, & inertia carere non posse, compenetrari tamen posse, vel 
replicari, vel sine inertia esse exiguas eorum partes. At si proprietas sit respectiva, respectu 
nostrorum sensuum, ex [19] eo, quod habeatur in majoribus massis, non debemus inferre, 
earn haberi in particulis minoribus, ut est hoc ipsum, esse sensibile, ut est, esse coloratas, 
quod ipsis majoribus massis competit, minoribus non competit ; cum ejusmodi magnitudinis 
discrimen, accidentale respectu materiae, non sit accidentale respectu ejus denominationis 
sensibile, coloratum. Sic etiam siqua proprietas ita pendet a ratione aggregati, vel totius, ut 
ab ea separari non possit ; nee ea, ob rationem nimirum eandem, a toto, vel aggregate debet 
transferri ad partes. Est de ratione totius, ut partes habeat, nee totum sine partibus haberi 
potest. Est de ratione figurabilis, & extensi, ut habeat aliquid, quod ab alio distet, adeoque, 
ut habeat partes ; hinc eae proprietates, licet in quovis aggregate particularum materiae, 
sive in quavis sensibili massa, inveniantur, non debent inductionis vi transferri ad particulas 

Et impenetrabili- 41. Ex his patet, & impenetrabilitatem, & continuitatis legem per ejusmodi inductionis 

ultatem tvtad""pCT genus abunde probari, atque evinci, & illam quidem ad quascunque utcunque exiguas 

inductionem : "* ad particulas corporum, hanc ad gradus utcunque exiguos momento temporis adjectos debere 

ipsam quid requu-a- exten( jj < Requiritur autem ad hujusmodi inductionem primo, ut ilia proprietas, ad quam 

probandam ea adhibetur, in plurimis casibus observetur, aliter enim probabilitas esset exigua ; 

& ut nullus sit casus observatus, in quo evinci possit, earn violari. Non est necessarium illud, 

ut in iis casibus, in quibus primo aspectu timeri possit defectus proprietatis ipsius, positive 

demonstretur, earn non deficere ; satis est, si pro iis casibus haberi possit ratio aliqua 

conciliandi observationem cum ipsa proprietate, & id multo magis, si in aliis casibus habeatur 

ejus conciliationis exemplum, & positive ostendi possit, eo ipso modo fieri aliquando 


Ejus appiicatio ad 42. Id ipsum fit, ubi per inductionem impenetrabilitas corporum accipitur pro generali 

impenetrab;htatem. j e g e ]sj aturaEi Nam impenetrabilitatem ipsam magnorum corporum observamus in exemplis 
sane innumeris tot corporum, quae pertractamus. Habentur quidem & casus, in quibus earn 
violari quis credent, ut ubi oleum per ligna, & marmora penetrat, atque insinuatur, & ubi 
lux per vitra, & gemmas traducitur. At praesto est conciliatio phasnomeni cum impenetra- 
bilitate, petita ab eo, quod ilia corpora, in quse se ejusmodi substantiae insinuant, poros 
habeant, quos 633 permeent. Et quidem haec conciliatio exemplum habet manifestissimum 
in spongia, quae per poros ingentes aqua immissa imbuitur. Poros marmorum illorum, & 
multo magis vitrorum, non videmus, ac multo minus videre possumus illud, non insinuari 
eas substantias nisi per poros. Hoc satis est reliquae inductionis vi, ut dicere debeamus, eo 
potissimum pacto se rem habere, & ne ibi quidem violari generalem utique impenetrabilitatis 

simiiisad continu- [ 2O ] 43- Eodem igitur pacto in lege ipsa continuitatis agendum est. Ilia tarn ampla 

itatem : duo cas- inductio, quam habemus, debet nos movere ad illam generaliter admittendam etiam pro iis 

quibus ea n< videatur casibus, in quibus determinare immediate per observations non possumus, an eadem 

lacdi - habeatur, uti est collisio corporum ; ac si sunt casus nonnulli, in quibus eadem prima fronte 

violari videatur ; ineunda est ratio aliqua, qua ipsum phsenomenum cum ea lege conciliari 

possit, uti revera potest. Nonnullos ejusmodi casus protuli in memoratis dissertationibus, 

quorum alii ad geometricam continuitatem pertinent, alii ad physicam. In illis prioribus 

non immorabor ; neque enim geometrica continuitas necessaria est ad hanc physicam 

propugnandam, sed earn ut exemplum quoddam ad confirmationem quandam inductionis 

majoris adhibui. Posterior, ut saepe & ilia prior, ad duas classes reducitur ; altera est eorum 

casuum, in quibus saltus videtur committi idcirco, quia nos per saltum omittimus intermedias 

quantitates : rem exemplo geometrico illustro, cui physicum adjicio. 


chosen rules. Now, since the indication may possibly be fallacious, it may happen that an 
error may be made ; but there is presumption against such an error, as they call it in law, 
until direct evidence to the contrary can be brought forward. Hence we should add : 
unless some positive argument is against it. Thus, it would be offending against these rules 
to say that large bodies indeed could not suffer compenetration, or enfolding, or be deficient 
in inertia, but yet very small parts of them could suffer penetration, or enfolding, or be 
without inertia. On the other hand, if a property is relative with respect to our senses, 
then, from a result obtained for the larger masses we cannot infer that the same is to be 
obtained in its smaller particles ; for instance, that it is the same thing to be sensible, as 
it is to be coloured, which is true in the case of large masses, but not in the case of small 
particles ; since a distinction of this kind, accidental with respect to matter, is not accidental 
with respect to the term sensible or coloured. So also if any property depends on an argu- 
ment referring to an aggregate, or a whole, in such a way that it cannot be considered 
apart from the whole, or the aggregate ; then, neither must it (that is to say, by that same 
argument), be transferred from the whole, or the aggregate, to parts of it. It is on account 
of its being a whole that it has parts ; nor can there be a whole without parts. It is on 
account of its being figurable & extended that it has some thing that is apart from some 
other thing, & therefore that it has parts. Hence those properties, although they are 
found in any aggregate of particles of matter, or in any sensible mass, must not however be 
transferred by the power of induction to each & every particle." 

41. From what has been said it is quite evident that both impenetrability & the Law Both impenetra- 
of Continuity can be proved by a kind of induction of this type ; & the former must be c^bf dTm""^ 
extended to all particles of bodies, no matter how small, & the latter to all additional steps, strated by indue- 
however small, made in an instant of time. Now, in the first place, to use this kind of quired for this S pur- 
induction, it is required that the property, for the proof of which it is to be used, must be pose. 
observed in a very large number of cases ; for otherwise the probability would be very 

small. Also it is required that no case should be observed, in which it can be proved that 
it is violated. It is not necessary that, in those cases in which at first sight it is feared that 
there may be a failure of the property, that it should be directly proved that there is no 
failure. It is sufficient if in those cases some reason can be obtained which will make the 
observation agree with the property ; & all the more so, if in other cases an example of 
reconciliation can be obtained, & it can be positively proved that sometimes reconciliation 
can be obtained in that way. 

42. This is just what does happen, when the impenetrability of solid bodies is accepted Application of in- 
as a law of Nature through inductive reasoning. For we observe this impenetrability of tr'abiuty.* lmpene " 
large bodies in innumerable examples of the many bodies that we consider. There are 

indeed also cases, in which one would think that it was violated, such as when oil penetrates 
wood and marble, & works its way through them, or when light passes through glasses & 
gems. But we have ready a means of making these phenomena agree with impenetrability, 
derived from the fact that those bodies, into which substances of this kind work their way, 
possess pores which they can permeate. There is a very evident example of this recon- 
ciliation in a sponge, which is saturated with water introduced into it by means of huge 
pores. We do not see the pores of the marble, still less those of glass ; & far less can we see 
that these substances do not penetrate except by pores. It satisfies the general force of 
induction if we can say that the matter can be explained in this way better than in any 
other, & that in this case there is absolutely no contradiction of the general law of impene- 

43. In the same way, then, we must deal with the Law of Continuity. The full Similar application 


induction that we possess should lead us to admit in general this law even in those cases in ^sisses "oT cases 

which it is impossible for us to determine directly by observation whether the same law which there seems 

holds good, as for instance in the collision of bodies. Also, if there are some cases in which * 

the law at first sight seems to be violated, some method must be followed, through which 

each phenomenon can be reconciled with the law, as is in every case possible. I brought 

forward several cases of this kind in the dissertations I have mentioned, some of which 

pertained to geometrical continuity, & others to physical continuity. I will not delay over 

the first of these : for geometrical continuity is not necessary for the defence of the physical 

variety ; I used it as an example in confirmation of a wider induction. The latter, as well 

as very frequently the former, reduces to two classes ; & the first of these classes is that class 

in which a sudden change seems to have been made on account of our having omitted the 

intermediate quantities with a jump. I give a geometrical illustration, and then add one 

in physics. 



Exemplum geome- 
tricum primi gene- 
ris, ubi nos inter- 
mcdias magnitu- 
dines omittimus. 

Quando id accidat 
exempla physica 
dierum, & oscilla- 
tionum consequen- 

44. In axe curvae cujusdam in fig. 4. sumantur segmenta AC, CE, EG aequalia, & 
erigantur ordinatae AB, CD, EF, GH. Area; BACD, DCEF, FEGH videntur continue 
cujusdam seriei termini ita, ut ab ilia BACD acl DCEF, & inde ad FEGH immediate 
transeatur, & tamen secunda a prima, ut 

& tertia a secunda, differunt per quanti- 
tates finitas : si enim capiantur CI, EK 
sequales BA, DC, & arcus BD transferatur 
in IK ; area DIKF erit incrementum se- 
cundae supra primam, quod videtur imme- 
diate advenire totum absque eo, quod 
unquam habitum sit ejus dimidium, vel 
quaevis alia pars incrementi ipsius ; ut idcirco 
a prima ad secundam magnitudinem areae 
itum sit sine transitu per intermedias. At 
ibi omittuntur a nobis termini intermedii, 

qui continuitatem servant ; si enim ac aequalis FIG. 4. 

AC motu continue feratur ita, ut incipiendo 

ab AC desinat in CE ; magnitude areae BACD per omnes intermedias bacd abit in magnitu- 
dinem DCEF sine ullo saltu, & sine ulla violatione continuitatis. 

45. Id sane ubique accidit, ubi initium secundae magnitudinis aliquo intervallo distal 
ab initio primas ; sive statim veniat post ejus finem, sive qua vis alia lege ab ea disjungatur. 
Sic in pliysicis, si diem concipiamus intervallum temporis ab occasu ad occasum, vel etiam 
ab ortu ad occasum, dies praecedens a sequent! quibusdam anni temporibus differt per plura 
secunda, ubi videtur fieri saltus sine ullo intermedio die, qui minus differat. At seriem 
quidem continuam ii dies nequaquam constituunt. Concipiatur parallelus integer Telluris, 
in quo sunt continuo ductu disposita loca omnia, quae eandem latitudinem geographicam 
habent ; ea singula loca suam habent durationem diei, & omnium ejusmodi dierum initia, 
ac fines continenter fluunt ; donee ad eundem redeatur locum, cujus pre-[2i]-cedens dies 
est in continua ilia serie primus, & sequens postremus. Illorum omnium dierum magni- 
tudines continenter fluunt sine ullo saltu : nos, intermediis omissis, saltum committimus 
non Natura. Atque huic similis responsio est ad omnes reliquos casus ejusmodi, in quibus 
initia, & fines continenter non fluunt, sed a nobis per saltum accipiuntur. Sic ubi pendulum 
oscillat in acre ; sequens oscillatio per finitam magnitudinem distat a praecedente ; sed & 
initium & finis ejus finite intervallo temporis distat a prascedentis initio, & fine, ac intermedii 
termini continua serie fluente a prima oscillatione ad secundam essent ii, qui haberentur, si 
primae, & secundae oscillationis arcu in aequalem partium numerum diviso, assumeretur via 
confecta, vel tempus in ea impensum, inter jacens inter fines partium omnium proportion- 
alium, ut inter trientem, vel quadrantem prioris arcus, & trientem,vel quadrantem posterioris, 
quod ad omnes ejus generis casus facile transferri potest, in quibus semper immediate etiam 
demonstrari potest illud, continuitatem nequaquam violari. 

Exempla secundi 46. Secunda classis casuum est ea, in qua videtur aliquid momento temporis peragi, 

at ne iOTime UtS sed & tamen peragitur tempore successive, sed perbrevi. Sunt, qui objiciant pro violatione 
non momento' tem- continuitatis casum, quo quisquam manu lapidem tenens, ipsi statim det velocitatem 
quandam finitam : alius objicit aquae e vase effluentis, foramine constitute aliquanto infra 
superficiem ipsius aquae, velocitatem oriri momento temporis finitam. At in priore casu 
admodum evidens est, momento temporis velocitatem finitam nequaquam produci. Tempore 
opus est, utcunque brevissimo, ad excursum spirituum per nervos, & musculos, ad fibrarum 
tensionem, & alia ejusmodi : ac idcirco ut velocitatem aliquam sensibilem demus lapidi, 
manum retrahimus, & ipsum aliquandiu, perpetuo accelerantes, retinemus. Sic etiam, ubi 
tormentum bellicum exploditur, videtur momento temporis emitti globus, ac totam 
celeritatem acquirere ; at id successive fieri, patet vel inde, quod debeat inflammari tota 
massa pulveris pyrii, & dilatari aer, ut elasticitate sua globum acceleret, quod quidem fit 
omnino per omnes gradus. Successionem multo etiam melius videmus in globe, qui ab 
elastro sibi relicto propellatur : quo elasticitas est major, eo citius, sed nunquam momento 
temporis velocitas in globum inducitur. 

AppUcatio ipsorum 47. Hsec exempla illud praestant, quod aqua per pores spongiae ingressa respectu 

ad emuxum 1I aquK impenetrabilitatis, ut ea responsione uti possimus in aliis casibus omnibus, in quibus accessio 
e vase. aliqua magnitudinis videtur fieri tota momento temporis ; ut nimirum dicamus fieri tempore 


44. In the axis of any curve (Fig. 4) let there be taken the segments AC, CE, EG equal Geometrical ex- 
to one another ; & let the ordinates AB, CD, EF, GH be erected. The areas BACD, DCEF, kind where 6 Ivl 
FEGH seem to be terms of some continuous series such that we can pass directly from BACD omit . intermediate 
to DCEF and then on to FEGH, & yet the second differs from the first, & also the third from 

the second, by a finite quantity. For if CI, EK are taken equal to BA, DC, & the arc BD 
is transferred to the position IK ; then the area DIKF will be the increment of the second 
area beyond the first ; & this seems to be directly arrived at as a whole without that which 
at any one time is considered to be the half of it, or indeed any other part of the increment 
itself : so that, in consequence, we go from the first to the second magnitude of area without 
passing through intermediate magnitudes. But in this case we omit intermediate terms 
which maintain the continuity ; for if ac is equal to AC, & this is carried by a continuous 
motion in such a way that, starting from the position AC it ends up at the position CE, 
then the magnitude of the area BACD will pass through all intermediate values such as 
bacd until it reaches the magnitude of the area DCEF without any sudden change, & hence 
without any breach of continuity. 

45. Indeed this always happens when the beginning of the second magnitude is distant when this will 
by a definite interval from the beginning of the first ; whether it comes immediately after ha ppen = physical 
the end of the first or is disconnected from it by some other law. Thus in physics, if we casTof Consecutive 
look upon the day as the interval of time between sunset & sunset, or even between sunrise da y^ OI . consecutive 
& sunset, the preceding day differs from that which follows it at certain times of the year 

by several seconds ; in which case we see that there is a sudden change made, without there 
being any intermediate day for which the change is less. But the fact is that these days do 
not constitute a continuous series. Let us consider a complete parallel of latitude on the 
Earth, along which in a continuous sequence are situated all those places that have the same 
geographical latitude. Each of these places has its own duration of the day, & the begin- 
nings & ends of days of this kind change uninterruptedly ; until we get back again to the 
same place, where the preceding day is the first of that continuous series, & the day that fol- 
lows is the last of the series. The magnitudes of all these days continuously alter without there 
being any sudden change : it was we who, by omitting the intermediates, made the sudden 
change, & not Nature. Similar to this is the answer to all the rest of the cases of the same 
kind, in which the beginnings & the ends do not change uninterruptedly, but are observed by 
us discontinuously. Similarly, when a pendulum oscillates in air, the oscillation that follows 
differs from the oscillation that has gone before by a finite magnitude. But both the begin- 
ning & the end of the second differs from the beginning & the end of the first by a finite inter- 
val of time ; & the intermediate terms in a continuously varying series from the first oscillation 
to the second would be those that would be obtained, if the arcs of the first & second oscilla- 
tions were each divided into the same number of equal parts, & the path traversed (or the 
time spent in traversing the path) is taken between the ends of all these proportional paths ; 
such as that between the third or fourth part of the first arc & the third or fourth part 
of the second arc. This argument can be easily transferred so as to apply to all cases of this 
kind ; & in such cases it can always be directly proved that there is no breach of continuity. 

46. The second class of cases is that in which something seems to have been done in an Examples of the 
instant of time, but still it is really done in a continuous, but very short, interval of time. ^iS? the^chan'e 
There are some who bring forward, as an objection in favour of a breach of continuity, the is very rapid, but 
case in which a man, holding a stone in his hand, gives to it a definite velocity all at once ; f^an^nstant^of 
another raises an objection that favours a breach of continuity, in the case of water flowing time. 

from a vessel, where, if an opening is made below the level of the surface of the water, a 
finite velocity is produced in an instant of time. But in the first case it is perfectly clear 
that a finite velocity is in no wise produced in an instant of time. For there is need of 
time, although this is exceedingly short, for the passage of cerebral impulses through 
the nerves and muscles, for the tension of the fibres, and other things of that sort ; and 
therefore, in order to give a definite sensible velocity to the stone, we draw back the hand, 
and then retain the stone in it for some time as we continually increase its velocity forwards. 
So too when an engine of war is exploded, the ball seems to be driven forth and to acquire 
the whole of its speed in an instant of time. But that it is done continuously is clear, if 
only from the fact that the whole mass of the gunpowder has to be inflamed and the gas 
has to be expanded in order that it may accelerate the ball by its elasticity ; and this latter 
certainly takes place by degrees. The continuous nature of this is far better seen in the 
case of a ball propelled by releasing a spring ; here the stronger the elasticity, the greater 
the speed ; but in no case is the speed imparted to the ball in an instant of time. 

47. These examples are superior to that of water entering through the pores of a sponge, Application of 
which we employed in the matter of impenetrability ; so that we can make use of this reply *.g s s . e particularly 
in all other cases in which some addition to a magnitude seems to have taken place entirely in to the flow of water 
an instant of time. Thus, without doubt we may say that it takes place in an exceedingly from a vesse1 ' 



brevissimo, utique per omnes intermedias magnitudines, ac illsesa penitus lege continuitatis. 
Hinc & in aquae effluentis exemplo res eodem redit, ut non unico momento, sed successive 
aliquo tempore, & per [22] omnes intermedias magnitudines progignatur velocitas, quod 
quidem ita se habere optimi quique Physici affirmant. Et ibi quidem, qui momento 
temporis omnem illam velocitatem progigni, contra me affirmet, principium utique, ut 
ajunt, petat, necesse est. Neque enim aqua, nisi foramen aperiatur, operculo dimoto, 
effluet ; remotio vero operculi, sive manu fiat, sive percussione aliqua, non potest fieri 
momento temporis, sed debet velocitatem suam acquirere per omnes gradus ; nisi illud 
ipsum, quod quaerimus, supponatur jam definitum, nimirum an in collisione corporum 
communicatio motus fiat momento temporis, an per omnes intermedios gradus, & magni- 
tudines. Verum eo omisso, si etiam concipiamus momento temporis impedimentum 
auferri, non idcirco momento itidem temporis omnis ilia velocitas produceretur ; ilia enim 
non a percussione aliqua, sed a pressione superincumbentis aquae orta, oriri utique non 
potest, nisi per accessiones continuas tempusculo admodum parvo, sed non omnino nullo : 
nam pressio tempore indiget, ut velocitatem progignat, in communi omnium sententia. 

Transitus ad meta- 


ut in Geometria. 

48. Illaesa igitur esse debet continuitatis lex, nee ad earn evertendam contra inductionem, 
tam uberem quidquam poterunt casus allati hucusque, vel iis similes. At ejusdem con- 
umcus, tinuitatis aliam metaphysicam rationem adinveni, & proposui in dissertatione De Lege 
Continuitatis, petitam ab ipsa continuitatis natura, in qua quod Aristoteles ipse olim 
notaverat, communis esse debet limes, qui praecedentia cum consequentibus conjungit, qui 
idcirco etiam indivisibilis est in ea ratione, in qua est limes. Sic superficies duo solida 
dirimens & crassitudine caret, & est unica, in qua immediatus ab una parte fit transitus ad 
aliam ; linea dirimens binas superficiei continuae partes latitudine caret ; punctum continuae 
lineae segmenta discriminans, dimensione omni : nee duo sunt puncta contigua, quorum 
alterum sit finis prioris segmenti, alterum initium sequentis, cum duo contigua indivisibilia, 
& inextensa haberi non possint sine compenetratione, & coalescentia quadam in unum. 

idem in tempore 49. Eodem autem pacto idem debet accidere etiam in tempore, ut nimirum inter tempus 

> ti ua^'evide" 6 contmuum praecedens, & continuo subsequens unicum habeatur momentum, quod sit 
tius in quibusdam. indivisibilis terminus utriusque ; nee duo momenta, uti supra innuimus, contigua esse 
possint, sed inter quodvis momentum, & aliud momentum debeat intercedere semper 
continuum aliquod tempus divisibile in infinitum. Et eodem pacto in quavis quantitate, 
quae continuo tempore duret, haberi debet series quasdam magnitudinum ejusmodi, ut 
momento temporis cuivis respondeat sua, quae praecedentem cum consequente conjungat, 
& ab ilia per aliquam determinatam magnitudinem differat. Quin immo in illo quantitatum 
genere, in quo [23] binae magnitudines simul haberi non possunt, id ipsum multo evidentius 
conficitur, nempe nullum haberi posse saltum immediatum ab una ad alteram. Nam illo 
momento temporis, quo deberet saltus fieri, & abrumpi series accessu aliquo momentaneo, 
deberent haberi duae magnitudines, postrema seriei praecedentis, & prima seriei sequentis. 
Id ipsum vero adhuc multo evidentius habetur in illis rerum statibus, in quibus ex una 
parte quovis momento haberi debet aliquis status ita, ut nunquam sine aliquo ejus generis 
statu res esse possit ; & ex alia duos simul ejusmodi status habere non potest. 

inde cur motus ip- r o> \& quidem satis patebit in ipso locali motu, in quo habetur phsenomenum omnibus 

calls non fiat, nisi > . . * , . r r . ...... \ ,. . , . . 

per Hneam contin- sane notissimum, sed cujus ratio non ita facile ahunde redditur, inde autem patentissima est, 
Corpus a quovis loco ad alium quemvis devenire utique potest motu continuo per lineas 
quascunque utcunque contortas, & in immensum productas quaquaversum, quae numero 
infinities infinitae sunt : sed omnino debet per continuam aliquam abire, & nullibi inter- 
ruptam. En inde rationem ejus rei admodum manifestam. Si alicubi linea motus abrum- 
peretur ; vel momentum temporis, quo esset in primo puncto posterioris lineae, esset 
posterius eo momento, quo esset in puncto postremo anterioris, vel esset idem, vel anterius ? 
In primo, & tertio casu inter ea momenta intercederet tempus aliquod continuum divisibile 
in infinitum per alia momenta intermedia, cum bina momenta temporis, in eo sensu accepta, 
in quo ego hie ea accipio, contigua esse non possint, uti superiusexposui. Quamobrem in 


short interval of time, and certainly passes through every intermediate magnitude, and that 
the Law of Continuity is not violated. Hence also in the case of water flowing from a 
vessel it reduces to the same example : so that the velocity is generated, not in a single 
instant, but in some continuous interval of time, and passes through all intermediate magni- 
tudes ; and indeed all the most noted physicists assert that this is what really happens. 
Also in this matter, should anyone assert in opposition to me that the whole of the speed 
is produced in an instant of time, then he must use a petitio principii, as they call it. For 
the water can-not flow out, unless the hole is opened, & the lid removed ; & the removal of 
the lid, whether done by hand or by a blow, cannot be effected in an instant of time, but 
must acquire its own velocity by degrees ; unless we suppose that the matter under investi- 
gation is already decided, that is to say, whether in collision of bodies communication of 
motion takes place in an instant of time or through all intermediate degrees and magnitudes. 
But even if that is left out of account, & if also we assume that the barrier is removed 
in an instant of time, none the more on that account would the whole of the velocity 
also be produced in an instant of time ; for it is impossible that such velocity can arise, 
not from some blow, but from a pressure arising from the superincumbent water, except by 
continuous additions in a very short interval of time, which is however not absolutely 
nothing ; for pressure requires time to produce velocity, according to the general opinion 
of everybody. 

48. The Law of Continuity ought then to be subject to no breach, nor will the cases Passing to a meta- 
hitherto brought forward, nor others like them, have any power at all to controvert this haveT'smrie'iinUt 
law in opposition to induction so copious. Moreover I discovered another argument, a in the case of con- 
metaphysical one, in favour of this continuity, & published it in my dissertation De Lege g'^n^iy 1 " 11 ^' &S "* 
Continuitatis, having derived it from the very nature of continuity ; as Aristotle himself long 

ago remarked, there must be a common boundary which joins the things that precede to 
those that follow ; & this must therefore be indivisible for the very reason that it is a 
boundary. In the same way, a surface of separation of two solids is also without thickness 
& is single, & in it there is immediate passage from one side to the other ; the line of 
separation of two parts of a continuous surface lacks any breadth ; a point determining 
segments of a continuous line has no dimension at all ; nor are there two contiguous points, 
one of which is the end of the first segment, & the other the beginning of the next ; for 
two contiguous indivisibles, of no extent, cannot possibly be considered to exist, unless 
there is compenetration & a coalescence into one. 

49. In the same way, this should also happen with regard to time, namely, that between similarly for time 
a preceding continuous time & the next following there should be a single instant, which ^ y . mor^evi- 
is the indivisible boundary of either. There cannot be two instants, as we intimated above, dent in some than 
contiguous to one another ; but between one instant & another there must always intervene m others - 

some interval of continuous time divisible indefinitely. In the same way, in any quantity 
which lasts for a continuous interval of time, there must be obtained a series of magnitudes 
of such a kind that to each instant of time there is its corresponding magnitude ; & this 
magnitude connects the one that precedes with the one that follows it, & differs from the 
former by some definite magnitude. Nay even in that class of quantities, in which we 
cannot have two magnitudes at the same time, this very point can be deduced far more 
clearly, namely, that there cannot be any sudden change from one to another. For at that 
instant, when the sudden change should take place, & the series be broken by some momen- 
tary definite addition, two magnitudes would necessarily be obtained, namely, the last of 
the first series & the first of the next. Now this very point is still more clearly seen in those 
states of things, in which on the one hand there must be at any instant some state so that 
at no time can the thing be without some state of the kind, whilst on the other hand it can 
never have two states of the kind simultaneously. 

50. The above will be sufficiently clear in the case of local motion, in regard to which Hence the reason 
the phenomenon is perfectly well known to all ; the reason for it, however, is not so easily ^Jj^ Recurs;:! 10 " 
derived from any other source, whilst it follows most clearly from this idea. A body can continuous line, 
get from any one position to any other position in any case by a continuous motion along 

any line whatever, no matter how contorted, or produced ever so far in any direction ; 
these lines being infinitely infinite in number. But it is bound to travel by some continuous 
line, with no break in it at any point. Here then is the reason of this phenomenon quite 
clearly explained. If the motion in the line should be broken at any point, either the 
instant of time, at which it was at the first point of the second part of the line, would be 
after the instant, at which it was at the last point of the first part of the line, or it would 
be the same instant, or before it. In the first & third cases, there would intervene between 
the two instants some definite interval of continuous time divisible indefinitely at other 
intermediate instants ; for two instants of time, considered in the sense in which I have 


primo casu in omnibus iis infinitis intermediis momentis nullibi esset id corpus, in secundo 
casu idem esset eodem illo memento in binis locis, adeoque replicaretur ; in terio haberetur 
replicatio non tantum respectu eorum binorum momentorum, sed omnium etiam inter- 
mediorum, in quibus nimirum omnibus id corpus esset in binis locis. Cum igitur corpus 
existens nee nullibi esse possit, nee simul in locis pluribus ; ilia vias mutatio, & ille saltus 
haberi omnino non possunt. 

51. Idem ope Geometric magis adhuc oculis ipsis subjicitur. Exponantur per rectam 
AB tempora, ac per ordinatas ad lineas CD, EF, abruptas alicubi, diversi status rei cujuspiam. 
e metaphysica, Ductis ordinatis DG, EH, vel punctum H iaceret post G, ut in Fie. c : vel cum ipso 

ibus exemphs . / i i j . r T . o J r 

congrueret, ut in 6 ; vel ipsum prsccederet, ut in 7. In pnmo casu nulla responderet 
ordinata omnibus punctis rectae GH ; in secundo binae responderent GD, & HE eidem puncto 
G ; in tertio vero binae HI, & HE puncto H, binas GD, GK puncto G, & binae LM, LN 

Illustratio ejus 
i ex Geo- 


pluribus exempl 

D E. 


G H 

FIG. 5. 

B A 


FIG. 6. 

H L G 

FIG. 7. 

puncto cuivis intermedio L ; nam ordinata est relatio quaedam distantly, quam habet 
punctum curvae cum puncto axis sibi respondente, adeoque ubi jacent in recta eadem 
perpendiculari axi bina curvarum puncta, habentur binae ordinatae respondentes eidem 
puncto axis. Quamobrem si nee o-[24]-mni statu carere res possit, nee haberi possint 
status simul bini ; necessario consequitur, saltum ilium committi non posse. Saltus ipse, si 
deberet accidere, uti vulgo fieri concipitur, accideret binis momentis G, & H, quae sibi in 
fig. 6 immediate succederent sine ullo immediato hiatu, quod utique fieri non potest ex 
ipsa limitis ratione, qui in continuis debet esse idem, & antecedentibus, & consequentibus 
communis, uti diximus. Atque idem in quavis reali serie accidit ; ut hie linea finita sine 
puncto primo, & postremo, quod sit ejus limes, & superficies sine linea esse non potest ; unde 
fit, ut in casu figurae 6 binae ordinatae necessario respondere debeant eidem puncto : ita in 
quavis finita reali serie statuum primus terminus, & postremus haberi necessario debent ; 
adeoque si saltus fit, uti supra de loco diximus ; debet eo momento, quo saltus confici 
dicitur, haberi simul status duplex ; qui cum haberi non possit : saltus itidem ille haberi 
omnino non potest. Sic, ut aliis utamur exemplis, distantia unius corporis ab alio mutari 
per saltum non potest, nee densitas, quia dux simul haberentur distantiae, vel duae densitates, 
quod utique sine replicatione haberi non potest ; caloris itidem, & frigoris mutatio in 
thermometris, ponderis atmosphaerae mutatio in barometris, non fit per saltum, quia binae 
simul altitudines mercurii in instrumento haberi deberent eodem momento temporis, quod 
fieri utique non potest ; cum quovis momento determinate unica altitude haberi debeat, 
ac unicus determinatus caloris gradus, vel frigoris ; quae quidem theoria innumeris casibus 
pariter aptari potest. 

52. Contra hoc argumentum videtur primo aspectu adesse aliquid, quod ipsum pforsus 
non esse conjun- evertat, & tamen ipsi illustrando idoneum est maxime. Videtur nimirum inde erui, 

gend s in creatione M o J 

& annihiiatione, ac impossibilem esse & creationem rei cujuspiam, Scintentum. 01 enim conjungendus est 
ejus soiutio. postremus terminus praecedentis seriei cum primo sequentis ;" in ipso transitu a non esse ad 

esse, vel vice versa, debebit utrumque conjungi, ac idem simul erit, & non erit, quod est 
absurdum. Responsio in promptu est. Seriei finita; realis, & existentis, reales itidem, & 
existentes termini esse debent ; non vero nihili, quod nullas proprietates habet, quas exigat, 
Hinc si realium statuum seriei altera series realium itidem statuum succedat, quae non 
sit communi termino conjuncta ; bini eodem momento debebuntur status, qui nimirum 
sint bini limites earundem. At quoniam non esse est merum nihilum ; ejusmodi series 
limitem nullum extremum requirit, sed per ipsum esse immediate, & directe excluditur. 
Quamobrem primo, & postremo momento temporis ejus continui, quo res est, erit utique, 
nee cum hoc esse suum non esse conjunget simul ; at si densitas certa per horam duret, turn 
momento temporis in aliam mutetur duplam, duraturam itidem per alteram sequentem 
horam ; momento temporis, [25] quod horas dirimit, binae debebunt esse densitates simul, 
nimirum & simplex, & dupla, quae sunt reales binarum realium serierum termini. 

Objectio ab esse, & 


considered them, cannot be contiguous, as I explained above. Wherefore in the first case, 
at all those infinite intermediate instants the body would be nowhere at all ; in the second 
case, it would be at the same instant in two different places & so there would be replication. 
In the third case, there would not only occur replication in respect of these two instants 
but for all those intermediate to them as well, in all of which the body would forsooth be 
in two places at the same time. Since then a body that exists can never be nowhere, nor 
in several places at one & the same time, there can certainly be no alteration of path & no 
sudden change. 

51. The same thing can be visualized better with the aid of Geometry. illustration of this 

Let times be represented by the straight line AB, & diverse states of any thing by SSyT^STS 
ordinates drawn to meet the lines CD, EF, which are discontinuous at some point. If the reasoning being 
ordmates DG, EH are drawn, either the point H will fall after the point G, as in Fig. 5 ; 
or it will coincide with it, as in Fig. 6 ; or it will fall before it, as in Fig. 7. In the first 
case, no ordinate will correspond to any one of the points of the straight line GH ; in the 
second case, GD and HE would correspond to the same point G ; in the third case, two 
ordinates, HI, HE, would correspond to the same point H, two, GD, GK, to the same 
point G, and two, LM, LN, to any intermediate point L. Now the ordinate is some relation 
as regards distance, which a point on the curve bears to the point on the axis that corresponds 
with it ; & thus, when two points of the curve lie in the same straight line perpendicular 
to the axis, we have two ordinates corresponding to the same point of the axis. Wherefore, 
if the thing in question can neither be without some state at each instant, nor is it possible 
that there should be two states at the same time, then it necessarily follows that the sudden 
change cannot be made. For this sudden change, if it is bound to happen, would take place 
at the two instants G & H, which immediately succeed the one the other without any direct 
gap between them ; this is quite impossible, from the very nature of a limit, which should 
be the same for,& common to, both the antecedents & the consequents in a continuous set, 
as has been said. The same thing happens in any series of real things ; as in this case there 
cannot be a finite line without a first & last point, each to be a boundary to it, neither can 
there be a surface without a line. Hence it comes about that in the case of Fig. 6 two 
ordinates must necessarily correspond to the same point. Thus, in any finite real series of 
states, there must of necessity be a first term & a last ; & so if a sudden change is made, as 
we said above with regard to position, there must be at the instant, at which the sudden 
change is said to be accomplished, a twofold state at one & the same time. Now since this 
can never happen, it follows that this sudden change is also quite impossible. Similarly, to 
make use of other illustrations, the distance of one body from another can never be altered 
suddenly, no more can its density ; for there would be at one & the same time two distances, 
or two densities, a thing which is quite impossible without replication. Again, the change 
of heat, or cold, in thermometers, the change in the weight of the air in barometers, does 
not happen suddenly ; for then there would necessarily be at one & the same time two 
different heights for the mercury in the instrument ; & this could not possibly be the case. 
For at any given instant there must be but one height, & but one definite degree of heat, 
& but one definite degree of cold ; & this argument can be applied just as well to innu- 
merable other cases. 

52. Against this argument it would seem at first sight that there is something ready to 
hand which overthrows it altogether ; whilst as a matter of fact it is peculiarly fitted to together of existence 
exemplify it. It seems that from this argument it follows that both the creation of any * non-existence a.t 

> i -11 rf >r T i < -i i the time of creation 

thing, & its destruction, are impossible, r or, it the last term of a series that precedes is to O r annihilation ; & 

be connected with the first term of the series that follows, then in the passage from a state its solution. 

of existence to one of non-existence, or vice versa, it will be necessary that the two are 

connected together ; & then at one & the same time the same thing will both exist & not 

exist, which is absurd. The answer to this is immediate. For the ends of a finite series 

that is real & existent must themselves be real & existent, not such as end up in absolute 

nothing, which has no properties. Hence, if to one series of real states there succeeds 

another series of real states also, which is not connected with it by a common term, then 

indeed there must be two states at the same instant, namely those which are their two 

limits. But since non-existence is mere nothing, a series of this kind requires no last limiting 

term, but is immediately & directly cut off by fact of existence. Wherefore, at the first & 

at the last instant of that continuous interval of time, during which the matter exists, it will 

certainly exist ; & its non-existence will not be connected with its existence simultaneously. 

On the other hand if a given density persists for an hour, & then is changed in an instant 

of time into another twice as great, which will last for another hour ; then in that instant 

of time which separates the two hours, there would have to be two densities at one & the 

same time, the simple & the double, & these are real terms of two real series. 



Unde hue transfer- 
enda solutio ipsa. 

Solutio petita ex 
geometrico exem- 


ex meta- 

Illustratio ulterior 

Applicatio ad crea- 
tionem, & annihi- 








m m* 












C E H H'E'C 7 

FIG. 8. 

53. Id ipsum in dissertatione De lege virium in Natura existentium satis, ni fallor, 
luculenter exposui, ac geometricis figuris illustravi, adjectis nonnullis, quae eodem recidunt, 
& quae in applicatione ad rem, de qua agimus, & in cujus gratiam haec omnia ad legem con- 
tinuitatis pertinentia allata sunt, proderunt infra ; libet autem novem ejus dissertationis 
numeros hue transferre integros, incipiendo ab octavo, sed numeros ipsos, ut & schematum 
numeros mutabo hie, ut cum superioribus consentiant. 

54. " Sit in fig. 8 circulus GMM'wz, qui referatur ad datam rectam AB per ordinatas 
HM ipsi rectae perpendiculares ; uti itidem perpendiculares sint binae tangentes EGF, 
E'G'F'. Concipiantur igitur recta quaedam indefinita ipsi rectse AB perpendicularis, motu 
quodam continuo delata ab A ad B. Ubi ea habuerit, positionem quamcumque GD, quae 
praecedat tangentem EF, vel C'D', quae consequatur tangentem E'F' ; ordinata ad circulum 
nulla erit, sive erit impossibilis, & ut Geometrae 

loquuntur, imaginaria. Ubicunque autem ea sit 

inter binas tangentes EGF, E'G'F', in HI, HT, 

occurret circulo in binis punctis M, m, vel M', m', 

& habebitur valor ordinate HM, HOT, vel H'M', 

H'm'. Ordinata quidem ipsa respondet soli inter- 

vallo EE' : & si ipsa linea AB referat tempus ; 

momentum E est limes inter tempus praecedens 

continuum AE, quo ordinata non est, & tempus 

continuum EE' subsequens, quo ordinata est ; punc- 

tum E' est limes inter tempus praecedens EE', quo 

ordinata est, & subsequens E'B, quo non est. Vita 

igitur quaedam ordinatae est tempus EE' ; ortus 

habetur in E, interitus in E'. Quid autem in 

ipso ortu, & interitu ? Habetur-ne quoddam esse 

ordinatas, an non esse ? Habetur utique esse, nimi- 

rum EG, vel E'G', non autem non esse. Oritur 

tota finitae magnitudinis ordinata EG, interit tota finite magnitudinis E'G', nee tamen 

ibi conjungit esse, & non esse, nee ullum absurdum secum trahit. Habetur momento E 

primus terminus seriei sequentis sine ultimo seriei praecedentis, & habetur momento E' 

ultimus terminus seriei praecedentis sine primo termino seriei sequentis." 

55. " Quare autem id ipsum accidat, si metaphysica consideratione rem perpendimus, 
statim patebit. Nimirum veri nihili nullae sunt verae proprietates : entis realis verae, & 
reales proprietates sunt. Quaevis realis series initium reale debet, & finem, sive primum, & 
ultimum terminum. Id, quod non est, nullam habet veram proprietatem, nee proinde sui 
generis ultimum terminum, aut primum exigit. Series praecedens ordinatae nullius, ultimum 
terminum non [26] habet, series consequens non habet primum : series realis contenta 
intervallo EE', & primum habere debet, & ultimum. Hujus reales termini terminum ilium 
nihili per se se excludunt, cum ipsum esse per se excludat non esse." 

56. " Atque id quidem manifestum fit magis : si consideremus seriem aliquam 
praecedentem realem, quam exprimant ordinatae ad lineam continuam PLg, quae respondeat 
toti tempori AE ita, ut cuivis momento C ejus temporis respondeat ordinata CL. Turn 
vero si momento E debeat fieri saltus ab ordinata Eg ad ordinatam EG : necessario ipsi 
momento E debent respondere binae ordinatae EG, Eg. Nam in tota linea PLg non potest 
deesse solum ultimum punctum g ; cum ipso sublato debeat adhuc ilia linea terminum 
habere suum, qui terminus esset itidem punctum : id vero punctum idcirco fuisset ante 
contiguum puncto g, quod est absurdum, ut in eadem dissertatione De Lege Continuitatis 
demonstravimus. Nam inter quodvis punctum, & aliud punctum linea aliqua interjacere 
debet ; quae si non inter jaceat ; jam ilia puncta in unicum coalescunt. Quare non potest 
deesse nisi lineola aliqua gL ita, ut terminus seriei praecedentis sit in aliquo momento C 
praecedente momentum E, & disjuncto ab eo per tempus quoddam continuum, in cujus 
temporis momentis omnibus ordi'nata sit nulla." 

57. " Patet igitur discrimen inter transitum a vero nihilo, nimirum a quantitate 
imaginaria, ad esse, & transitum ab una magnitudine ad aliam. In primo casu terminus 
nihili non habetur ; habetur terminus uterque seriei veram habentis existentiam, & potest 
quantitas, cujus ea est series, oriri, vel occidere quantitate finita, ac per se excludere non esse. 
In secundo casu necessario haberi debet utriusque seriei terminus, alterius nimirum postre- 
mus, alterius primus. Quamobrem etiam in creatione, & in annihilatione potest quantitas 
oriri, vel interire magnitudine finita, & primum, ac ultimum esse erit quoddam esse, quod 
secum non conjunget una non esse. Contra vero ubi magnitude realis ab una quantitate ad 


c*. I explained this very point clearly enough, if I mistake not, in my dissertation The s " 166 from 

n i JIT- . ' . a T -11 j v i ... i A 'IT ^ which the solution 

D,? lege vmum in Natura existentium, & 1 illustrated it by geometrical figures ; also I made u to be borrowed. 

some additions that reduced to the same thing. These will appear below, as an application 

to the matter in question ; for the sake of which all these things relating to the Law of 

Continuity have been adduced. It is allowable for me to quote in this connection the 

whole of nine articles from that dissertation, beginning with Art. 8 ; but I will here 

change the numbering of the articles, & of the diagrams as well, so that they may agree 

with those already given. 

54. " In Fig. 8, let GMM'm be a circle, referred to a given straight line AB as axis, by Sotoion derived 
means of ordinates HM drawn perpendicular to that straight line ; also let the two tan- exampief" 
gents EGF, E'G'F' be perpendiculars to the axis. Now suppose that an unlimited straight 

line perpendicular to the axis AB is carried with a continuous motion from A to B. When 
it reaches some such position as CD preceding the tangent EF, or as C'D' subsequent to 
the tangent E'F', there will be no ordinate to the circle, or it will be impossible &, as the 
geometricians call it, imaginary. Also, wherever it falls between the two tangents EGF, 
E'G'F', as at HI or HT, it will meet the circle in two points, M, m or M', m' ; & for the 
value of the ordinate there will be obtained HM & Hm, or H'M' & H'm'. Such an ordinate 
will correspond to the interval EE' only ; & if the line AB represents time, the instant E 
is the boundary between the preceding continuous time AE, in which the ordinate does 
not exist, the subsequent continuous time EE', in which the ordinate does exist. The 
point E' is the boundary between the preceding time EE', in which the ordinate does exist, 
& the subsequent time E'B, in which it does not ; the lifetime, as it were, of the ordinate, 
is EE' ; its production is at E & its destruction at E'. But what happens at this production 
& destruction ? Is it an existence of the ordinate, or a non-existence I Of a truth there 
is an existence, represented by EG & E'G', & not a non-existence. The whole ordinate EG 
of finite magnitude is produced, & the whole ordinate E'G' of finite magnitude is destroyed; 
& yet there is no connecting together of the states of existence & non-existence, nor does it 
bring in anything absurd in its train. At the instant E we get the first term of the sub- 
sequent series without the last term of the preceding series ; & at the instant E' we have 
the last term of the preceding series without the first term of the subsequent series." 

55. " The reason why this should happen is immediately evident, if we consider the Sol tion from a 
matter metaphysically. Thus, to absolute nothing there belong no real properties ; but Sderatwn!* 

the properties of a real absolute entity are also real. Any real series must have a real 
beginning & end, or a first term & a last. That which does not exist can have no true 
property ; & on that account does not require a last term of its kind, or a first. The 
preceding series, in which there is no ordinate, does not have a last term ; & the subsequent 
series has likewise no first term ; whilst the real series contained within the interval EE' 
must have both a first term & a last term. The real terms of this series of themselves 
exclude the term of no value, since the fact of existence of itself excludes non-existence" 

56. " This indeed will be still more evident, if we consider some preceding series of Further illustration 

i 11 i i i i T.T r i i by geometry. 

real quantities, expressed by the ordinates to the curved line PLg ; & let this curve 
correspond to the whole time AE in such a way that to every instant C of the time there 
corresponds an ordinate CL. Then, if at the instant E there is bound to be a sudden 
change from the ordinate Eg to the ordinate EG, to that instant E there must of necessity 
correspond both the ordinates EG, Eg. For it is impossible that in the whole line PLg 
the last point alone should be missing ; because, if that point is taken away, yet the line 
is Bound to have an end to it, & that end must also be a point ; hence that point would be 
before & contiguous to the point g ; & this is absurd, as we have shown in the same 
dissertation De Lege Continuitatis. For between any one point & any other point there 
must lie some line ; & if such a line does not intervene, then those points must coalesce 
into one. Hence nothing can be absent, except it be a short length of line gL, so that 
the end of the series that precedes occurs at some instant, C, preceding the instant E, & 
separated from it by an interval of continuous time, at all instants of which there is no 

157. "Evidently, then, there is a distinction between passing from absolute nothing, Application to crea- 

f' . '' . ... . , . Y tion& annihilation. 

i.e., from an imaginary quantity, to a state of existence, & passing from one magnitude 
to another. In the first case the term which is naught is not reckoned in ; the term at 
either end of a series which has real existence is given, & the quantity, of which it is the 
series, can be produced or destroyed, finite in amount ; & of itself it will exclude non- 
existence. In the second case, there must of necessity be an end to either series, namely 
the last of the one series & the first of the other. Hence, in creation & annihilation, 
a quantity can be produced or destroyed, finite in magnitude ; & the first & last 
state of existence will be a state of existence of some kind ; & this will not associate with 
itself a state of non-existence. But, on the other hand, where a real magnitude is bound 



Aliquando videri 
nihtium id, quod 
est aliquid. 

Ordinatam nullam, 
ut & distantiam 
nullam existentium 
esse compenetra- 

Ad idem pertinere 
seriei realis genus 
earn distan t i a m 
nullam, & aliquam. 

Alia, quje videntur 
nihil, & sunt ali- 
quid : discrimen 
inter radicem ima- 
ginariam, & zero. 

aliam transire debet per saltum ; momento temporis, quo saltus committitur, uterque 
terminus haberi deberet. Manet igitur illaesum argumentum nostrum metaphysicum pro 
exclusione saltus a creatione & annihilatione, sive ortu, & interitu." 

58. "At hie illud etiam notandum est ; quoniam ad ortum, & interitum considerandum 
geometricas contemplationes assumpsimus, videri quidem prima fronte, aliquando etiam 
realis seriei terminum postremum esse nihilum ; sed re altius considerata, non erit vere 
nihilum ; sed status quidam itidem realis, & ejusdem generis cum prsecedentibus, licet alio 
nomine insignitus." 

[27] 59. " Sit in Fig. 9. Linea AB, ut prius, ad quam linea qusedam PL deveniat in G 
(pertinet punctum G ad lineam PL, E ad AB continuatas, & sibi occurrentes ibidem), & sive 
pergat ultra ipsam in GM', sive retro resiliat per GM'. Recta CD habebit ordinatam CL, 
quas evanescet, ubi puncto C abeunte in E, ipsa CD abibit in EF, turn in positione ulteriori 
rectse perpendicularis HI, vel abibit in nega- 
tivam HM, vel retro positiva regredietur 
in HM'. Ubi linea altera cum altera coit, 
& punctum E alterius cum alterius puncto 
G congreditur, ordinata CL videtur abire in 
nihilum ita, ut nihilum, quemadmodum & 
supra innuimus, sit limes quidam inter seriem 
ordinatarum positivarum CL, & negativarum 
HM ; vel positivarum CL, & iterum posi- 
tivarum HM'. Sed, si res altius considere- 
tur ad metaphysicum conceptum reducta, 
in situ EF non habetur verum nihilum. 
In situ CD, HI habetur distantia quaedam 
punctorum C, L ; H, M : in situ EF 
habetur eorundem punctorum compene- 

tratio. Distantia est relatio quaedam FJG 

binorum modorum, quibus bina puncta 

existunt ; compenetratio itidem est relatio binorum modorum, quibus ea existunt, 
quae compenetratio est aliquid reale ejusdem prorsus generis, cujus est distantia, constituta 
nimirum per binos reales existendi modos." 

60. " Totum discrimen est in vocabulis, quae nos imposuimus. Bini locales existendi 
modi infinitas numero relationes possunt constituere, alii alias. Hae omnes inter se & 
differunt, & tamen simul etiam plurimum conveniunt ; nam reales sunt, & in quodam genere 
congruunt, quod nimirum sint relationes ortae a binis localibus existendi modis. Diversa 
vero habent nomina ad arbitrarium instituta, cum alise ex ejusmodi relationibus, ut CL, 
dicantur distantiae positivae, relatio EG dicatur compenetratio, relationes HM dicantur 
distantiae negativse. Sed quoniam, ut a decem palmis distantiae demptis 5, relinquuntur 5, 
ita demptis aliis 5, habetur nihil (non quidem verum nihil, sed nihil in ratione distantiae a 
nobis ita appellatae, cum remaneat compenetratio) ; ablatis autem aliis quinque, remanent 
quinque palmi distantiae negativae ; ista omnia realia sunt, & ad idem genus pertinent ; cum 
eodem prorsus modo inter se differant distantia palmorum 10 a distantia palmorum 5, haec 
a distantia nulla, sed reali, quas compenetrationem importat, & haec a distantia negativa 
palmorum 5. Nam ex prima ilia quantitate eodem modo devenitur ad hasce posteriores per 
continuam ablationem palmorum 5. Eodem autem pacto infinitas ellipses, ab infinitis 
hyperbolis unica interjecta parabola discriminat, quae quidem unica nomen peculiare sortita 
est, cum illas numero infinitas, & a se invicem admodum discrepantes unico vocabulo com- 
plectamur ; licet altera magis oblonga ab altera minus oblonga plurimum itidem diversa sit." 

[28] 61. " Et quidem eodem pacto status quidam realis est quies, sive perseverantia in 
eodem modo locali existendi ; status quidam realis est velocitas nulla puncti existentis. 
nimirum determinatio perseverandi in eodem loco ; status quidam realis puncti existentis 
est vis nulla, nimirum determinatio retinendi praecedentem velocitatem, & ita porro ; 
plurimum haec discrepant a vero non esse. Casus ordinatae respondentis lineae EF in fig. 9, 
differt plurimum a casu ordinatae circuli respondentis lineae CD figurae 8 : in prima existunt 
puncta, sed compenetrata, in secunda alterum punctum impossible est. Ubi in solutione 
problematum devenitur ad quantitatem primi generis, problema determinationem peculiarem 
accipit ; ubi devenitur ad quantitatem secundi generis, problema evadit impossible ; usque 
adeo in hoc secundo casu habetur verum nihilum, omni reali proprietate carens ; in illo 
primo habetur aliquid realibus proprietatibus praeditum, quod ipsis etiam solutionibus 
problematum, & constructionibus veras sufficit, & reales determinationes ; cum realis, non 
imaginaria sit radix equationis cujuspiam, quae sit = o, sive nihilo aequalis." 


to pass suddenly from one quantity to another, then at the instant in which the sudden 
change is accomplished, both terms must be obtained. Hence, our argument on 
metaphysical grounds in favour of the exclusion of a sudden change from creation or 
annihilation, or production & destruction, remains quite unimpaired." 

58. " In this connection the following point must be noted. As we have used geometrical Sometimes what is 
ideas for the consideration of production & destruction, it seems also that sometimes reall y some thingap- 
the last term of a real series is nothing. But if we go deeper into the matter, we find 

that it is not in reality nothing, but some state that is also real and of the same kind as 
those that precede it, though designated by another name." 

59. " In Fig. 9, let AB be a line, as before, which some line PL reaches at G (where the When the ordinate 
point G belongs to the line PL, & E to the line AB, both being produced to meet one whe^thT'dlst^n' 13 
another at this point) ; & suppose that PL either goes on beyond the point as GM, or between two exis- 
recoils along GM'. Then the straight line CD will contain the ordinate CL, which will ^ t J 1 gs . u no " 

_ & , . . . ' . thing, there is com- 

vanish when, as the point L, gets to H, L-D attains the position r,r ; & after that, in the penetration. 

further position of the perpendicular straight line HI, will either pass on to the negative 

ordinate HM or return, once more positive, to HM'. Now when the one line meets the 

other, & the point E of the one coincides with the point G of the other, the ordinate 

CL seems to run off into nothing in such a manner that nothing, as we remarked above, 

is a certain boundary between the series of positive ordinates CL & the negative ordinates 

HM, or between the positive ordinates CL & the ordinates HM' which are also positive. 

But if the matter is more deeply considered & reduced to a metaphysical concept, there 

is not an absolute nothing in the position EF. In the position CD, or HI, we have given 

a certain distance between the points C,L, or H,M ; in the position EF, there is 

compenetration of these points. Now distance is a relation between the modes of existence 

of two points ; also compenetration is a relation between two modes of existence ; & 

this compenetration is something real of the very same nature as distance, founded as it is 

on two real modes of existence." 

60. " The whole difference lies in the words that we have given to the things in question. s ' no ' distance 
Two local modes of existence can constitute an infinite number of relations, some of one kmdT^f series "of 
sort & some of another. All of these differ from one another, & yet agree with one real quantities as 

i i i j r ia j i j j ' some ' distance. 

another in a high degree ; ior they are real & to a certain extent identical, since indeed 
they are all relations arising from a pair of local modes of existence. But they have different 
names assigned to them arbitrarily, so that some of the relations of this kind, as CL, are 
called positive distances, the relation EG is called compenetration, & relations like HM 
are called negative distances. But, just as when five palms of distance are taken away 
from ten palms, there are left five palms, so when five more are taken away, there is nothing 
left (& yet not really nothing, but nothing in comparison with what we usually call 
distance ; for compenetration is left). Again, if we take away another five, there remain 
five palms of negative distance. All of these are real & belong to the same class ; for 
they differ amongst themselves in exactly the same way, namely, the distance of ten palms 
from the distance of five palms, the latter from ' no ' distance (which however is something 
real that denotes compenetration), & this again from a negative distance of five palms. 
For starting with the first quantity, the others that follow are obtained in the same manner, 
by a continual subtraction of five palms. In a similar manner a single intermediate 
parabola discriminates between an infinite number of ellipses & an infinite number of 
hyperbolas ; & this single curve receives a special name, whilst under the one term we include 
an infinite number of them that to a certain extent are all different from one another, 
although one that is considerably elongated may be very different from another that is 
less elongated." 

61. "In the same way, rest, i.e., a perseverance in the same mode of local existence, other things that 
is some real state ; so is ' no ' velocity a real state of an existent point, namely, a propensity ^ndVet^re^eaJi^ 
to remain in the same place ; so also is ' no ' force a real state of an existent point, namely, something ; d i s- 
a propensity to retain the velocity that it has already; & so on. All these differ from a '~" 

a state of non-existence in the highest degree. The case of the ordinate corresponding & zero/ 
to the line EF in Fig. 9 differs altogether from the case of the ordinate of the circle 
corresponding to the line CD in Fig. 8. In the first there exist two points, but there is 
compenetration of these points ; in the other case, the second point cannot possibly exist. 
When, in the solution of problems, we arrive at a quantity of the first kind, the problem 
receives a special sort of solution ; but when the result is a quantity of the second kind, 
the problem turns out to be incapable of solution. So much indeed that, in this second case, 
there is obtained a true nothing that lacks every real property ; in the first case, we get 
something endowed with real properties, which also supplies true & real values to the 
solutions & constructions of the problems. For the root of any equation that = o, or is 
equal to nothing, is something that is real, & is not an imaginary thing." 


Conciusip prosolu- fa. " Firmum igitur manebit semper. & stabile, seriem realem quamcunque. quas 

tione ejus objec- . ~ . , , , v ... a i r 

contmuo tempore finito duret, debere habere primum prmcipium, & ultimum nnem 
realem, sine ullo absurdo, & sine conjunctione sui esse cum non esse, si forte duret eo solo 
tempore : dum si prascedenti etiam exstitit tempore, habere debet & ultimum terminum 
seriei praecedentis, & primum sequentis, qui debent esse unicus indivisibilis communis limes, 
ut momentum est unicus indivisibilis limes inter tempus continuum praecedens, & subsequens. 
Sed haec de ortu, & interitu jam satis." 

Apphcatio leg is ft- ij t igitur contrahamus iam vela, continuitatis lex & inductione, & metaphysico 

contmuitatis ad J , , . . i . .. . . r ' . 

coiiisionem corpo- argumento abunde nititur, quas idcirco etiam in velocitatis commumcatione retmeri omnmo 
rum - debet, ut nimirum ab una velocitate ad aliam numquam transeatur, nisi per intermedias 

velocitates omnes sine saltu. Et quidem in ipsis motibus, & velocitatibus inductionem 
habuimus num. 39, ac difficultates solvimus num. 46, & 47 pertinentes ad velocitates, quae 
videri possent mutatse per saltum. Quod autem pertinet ad metaphysicum argumentum, si 
toto tempore ante contactum subsequentis corporis superficies antecedens habuit 12 gradus 
velocitatis, & sequenti 9, saltu facto momentaneo ipso initio contactus ; in ipso momento ea 
tempora dirimente debuisset habere & 12, & 9 simul, quod est absurdum. Duas enim 
velocitates simul habere corpus non potest, quod ipsum aliquanto diligentius demonstrabo. 

DUO velocitatum g, Velocitatis nomen, uti passim usurpatur a Mechanicis, asquivocum est; potest 

genera, potentials, T r r . T. . r 

& actuaiis. enim sigmncare velocitatem actuaiem, quas nimirum est relatio quaedam in motu asquabm 

spatii percursi divisi per tempus, quo percurritur ; & potest significare [29] quandam, quam 
apto Scholiasticorum vocabulo potentialem appello, quae nimirum est determinatio, ad 
actuaiem, sive determinatio, quam habet mobile, si nulla vis mutationem inducat, percur- 
rendi motu asquabili determinatum quoddam spatium quovis determinato tempore, quas 
quidem duo & in dissertatione De Viribus Fivis, & in Stayanis Supplements distinxi, 
distinctione utique .necessaria ad aequivocationes evitandas. Prima haberi non potest 
momento temporis, sed requirit tempus continuum, quo motus fiat, & quidem etiam motum 
aequabilem requirit ad accuratam sui mensuram ; secunda habetur etiam momento quovis 
determinata ; & hanc alteram intelligunt utique Mechanici, cum scalas geometricas effor- 
mant pro motibus quibuscunque difformibus, sive abscissa exprimente tempus, & ordinata 
velocitatem, utcunque etiam variatam, area exprimat spatium : sive abscissa exprimente 
itidem tempus, & ordinata vim, area exprimat velocitatem jam genitam, quod itidem in aliis 
ejusmodi scalis, & formulis algebraicis fit passim, hac potentiali velocitate usurpata, quas sit 
tantummodo determinatio ad actuaiem, quam quidem ipsam intelligo, ubi in collisione 
corporum earn nego mutari posse per saltum ex hoc posteriore argumento. 

^5' J am vero velocitates actuales non posse simul esse duas in eodem mobili, satis patet ; 
potentials 'simul quia oporteret, id mobile, quod initio dati cujusdam temporis fuerit in dato spatii puncto, 
ne^etur n< vei exf<*a- ^ n omn ibus sequentibus occupare duo puncta ejusdem spatii, ut nimirum spatium percursum 
tur compenetratfo. sit duplex, alterum pro altera velocitate determinanda, adeoque requireretur actuaiis 
replicatio, quam non haberi uspiam, ex principio inductionis colligere sane possumus 
admodum facile. Cum nimirum nunquam videamus idem mobile simul ex eodem loco 
discedere in partes duas, & esse simul in duobis locis ita, ut constet nobis, utrobique esse illud 
idem. At nee potentiales velocitates duas simul esse posse, facile demonstratur. Nam 
velocitas potentialis est determinatio ad existendum post datum tempus continuum quodvis 
in dato quodam puncto spatii habente datam distantiam a puncto spatii, in quo mobile est 
eo temporis momento, quo dicitur habere illam potentialem velocitatem determinatam. 
Quamobrem habere simul illas duas potentiales velocitates est esse determinatum ad occu- 
panda eodem momento temporis duo puncta spatii, quorum singula habeant suam diversam 
distantiam ab eo puncto spatii, in quo turn est mobile, quod est esse determinatum ad 
replicationem habendam momentis omnibus sequentis temporis. Dicitur utique idem 
mobile a diversis causis acquirere simul diversas velocitates, sed eae componuntur in unicam 
ita, ut singulas constituant statum mobilis, qui status respectu dispositionum, quas eo 
momento, in quo turn est, habet ipsum mobile, complectentium omnes circumstantias 
praeteritas, & praesentes, est tantummodo conditionatus, non absolutus ; nimirum ut con- 
tineant determi-[3o]-nationem, quam ex omnibus praeteritis, & praesentibus circumstantiis 
haberet ad occupandum illud determinatum spatii punctum determinato illo momento 


62. "Hence in all cases it must remain a firm &stable conclusion that any real series, Conclusion in 

,.,, , c . . . .1 i, c i r- i favour of a solution 

which lasts for some finite continuous time, is bound to have a first beginning & a final O f this difficulty. 

end, without any absurdity coming in, & without any linking up of its existence with 

a state of non-existence, if perchance it lasts for that interval of time only. But if it existed 

at a previous time as well, it must have both a last term of the preceding series & a first 

term of the subsequent series ; just as an instant is a single indivisible boundary between 

the continuous time that precedes & that which follows. But what I have said about 

production & destruction is already quite enough." 

63. But, to come back at last to our point, the Law of Continuity is solidly founded Application of the 
both on induction & on metaphysical reasoning ; & on that account it should be retained ^The* co5ision"af 
in every case of communication of velocity. So that indeed there can never be any passing solid bodies. 
from one velocity to another except through all intermediate velocities, & then without 

any sudden change. We have employed induction for actual motions & velocities in 
Art. 39 & solved difficulties with regard to velocities in Art. 46, 47, in cases in which they 
might seem to be subject to sudden changes. As regards metaphysical argument, if in the 
whole time before contact the anterior surface of the body that follows had 12 degrees of 
velocity & in the subsequent time had 9, a sudden change being made at the instant of first 
contact ; then at the instant that separates the two times, the body would be bound to have 
12 degrees of velocity, & 9, at one & the same time. This is absurd ; for a body cannot at 
the same time have two velocities, as I will now demonstrate somewhat more carefully. 

64. The term velocity, as it is used in general by Mechanicians is equivocal. For it Two kinds of veio- 
may mean actual velocity, that is to say, a certain relation in uniform motion given by Clty< P tentlal & 
the space passed over divided by the time taken to traverse it. It may mean also something 

which, adopting a term used by the Scholastics, I call potential velocity. The latter is 
a propensity for actual velocity, or a propensity possessed by the movable body (should 
no force cause an alteration) for traversing with uniform motion some definite space in 
any definite time. I made the distinction between these two meanings, both in the 
dissertation De Firibus Fivis & in the Supplements to Stay's Philosophy ; the distinction 
being very necessary to avoid equivocations. The former cannot be obtained in an instant 
of time, but requires continuous time for the motion to take place ; it also requires uniform 
motion in order to measure it accurately. The latter can be determined at any given 
instant ; & it is this kind that is everywhere intended by Mechanicians, when they make 
geometrical measured diagrams for any non-uniform velocities whatever. In which, if 
the abscissa represents time & the ordinate velocity, no matter how it is varied, then 
the area will express the distance passed over ; or again, if the abscissa represents time 
& the ordinate force, then the area will represent the velocity already produced. This 
is always the case, for other scales of the same kind, whenever algebraical formulae & 
this potential velocity are employed ; the latter being taken to be but the propensity for 
actual velocity, such indeed as I understand it to be, when in collision of bodies I deny 
from the foregoing argument that there can be any sudden change. 

65. Now it is quite clear that there cannot be two actual velocities at one & the same I4 is impossible 
time in the same moving body. For, then it would be necessary that the moving body, have two velocities" 
which at the beginning of a certain time occupied a certain given point of space, should at either actual or 
all times afterwards occupy two points of that space ; so that the space traversed would be ^given) or we are 
twofold, the one space being determined by the one velocity & the other by the other, forced to admit, 
Thus an actual replication would be required ; & this we can clearly prove in a perfectly penetration ' S 
simple way from the principle of induction. Because, for instance, we never see the same 

movable body departing from the same place in two directions, nor being in two places at 
the same time in such a way that it is clear to us that it is in both. Again, it can be easily 
proved that it is also impossible that there should be two potential velocities at the same 
time. For potential velocity is the propensity that the body has, at the end of any given 
continuous time, for existing at a certain given point of space that has a given distance 
from that point of space, which the moving body occupied at the instant of time in which 
it is said to have the prescribed potential velocity. Wherefore to have at one & the same 
time two potential velocities is the same thing as being prescribed to occupy at the same 
instant of time two points of space ; each of which has its own distinct distance from that 
point of space that the body occupied at the start ; & this is the same thing as prescribing 
that there should be replication at all subsequent instants of time. It is commonly said 
that a movable body acquires from different causes several velocities simultaneously ; but 
these velocities are compounded into one in such a way that each produces a state of the 
moving body ; & this state, with regard to the dispositions that it has at that instant (these 
include all circumstances both past & present), is only conditional, not absolute. That is 
to say, each involves the propensity which the body, on account of all past & present 
circumstances, would have for occupying that prescribed point of space at that particular 


temporis ; nisi aliunde ejusmodi determinatio per conjunctionem alterius causae, quae turn 
agat, vel jam egerit, mutaretur, & loco ipsius alia, quae composita dicitur, succederet. Sed 
status absolutus resultans ex omnibus eo momento praasentibus, & prseteritis circumstantiis 
ipsius mobilis, est unica determinatio ad existendum pro quovis determinato momento 
temporis sequentis in quodam determinato puncto spatii, qui quidem status pro circum- 
stantiis omnibus praeteritis, & prsesentibus est absolutus, licet sit itidem conditionatus pro 
futuris : si nimirum esedem, vel alias causa; agentes sequentibus momentis non mutent 
determinationem, & punctum illud loci, ad quod revera deveniri deinde debet dato illo 
momento temporis, & actu devenitur ; si ipsae nihil aliud agant. Porro patet ejusmodi 
status ex omnibus prseteritis, & praesentibus circumstantiis absolutes non posse eodem 
momento temporis esse duos sine determinatione ad replicationem, quam ille conditionatus 
status resultans e singulis componentibus velocitatibus non inducit ob id ipsum, quod 
conditionatus est. Jam vero si haberetur saltus a velocitate ex omnibus prsteritis, & 
praesentibus circumstantiis exigente, ex. gr. post unum minutum, punctum spatii distans 
per palmos 6 ad exigentem punctum distans per palmos 9 ; deberet eo momento temporis, 
quo fieret saltus, haberi simul utraque determinatio absoluta respectu circumstantiarum 
omnium ejus momenti, & omnium praeteritarum ; nam toto prsecedenti tempore habita 
fuisset realis series statuum cum ilia priore, & toto sequenti deberet haberi cum ilia 
posteriore, adeoque eo momento, simul utraque, cum neutra series realis sine reali suo 
termino stare possit. 

Quovis momento 66. Praeterea corporis, vel puncti existentis potest utique nulla esse velocitas actualis, 

de n be U re hTbeTe saltern accurate talis ; si nimirum difformem habeat motum, quod ipsum etiam semper in 

statum reaiem ex Natura accidit, ut demonstrari posse arbitror, sed hue non pertinet ; at semper utique 

potentialis' 6 li!itlS haberi debet aliqua velocitas potentialis, vel saltern aliquis status, qui licet alio vocabulo 

appellari soleat, & dici velocitas nulla, est tamen non nihilum quoddam, sed realis status, 

nimirum determinatio ad quietem, quanquam hanc ipsam, ut & quietem, ego quidem 

arbitrer in Natura reapse haberi nullam, argumentis, quae in Stayanis Supplementis exposui 

in binis paragraphis de spatio, ac tempore, quos hie addam in fine inter nonnulla, quae hie 

etiam supplementa appellabo, & occurrent primo, ac secundo loco. Sed id ipsum itidem 

nequaquam hue pertinet. lis etiam penitus praetermissis, eruitur e reliquis, quae diximus, 

admisso etiam ut existente, vel possibili in Natura motu uniformi, & quiete, utramque 

velocitatem habere conditiones necessarias ad [31] hoc, ut secundum argumentum pro 

continuitatis lege superius allatum vim habeat suam, nee ab una velocitate ad alteram abiri 

possit sine transitu per intermedias. 

ento te^oris'trari" ^7' P atet auteni j nmc illud evinci, nee interire momento temporis posse, nee oriri 

sin ab una veioci- velocitatem totam corporis, vel puncti non simul intereuntis, vel orientis, nee hue transferri 
demonstrat liai & P osse quod de creatione, & morte diximus ; cum nimirum ipsa velocitas nulla corporis, vel 
vindicatur. puncti existentis, sit non purum nihil, ut monui, sed realis quidam status, qui simul cum 

alio reali statu determinatae illius intereuntis, vel orientis velocitatis deberet conjungi ; unde 
etiam fit, ut nullum effugium haberi possit contra superiora argumenta, dicendo, quando a 
12 gradibus velocitatis transitur ad 9, durare utique priores 9, & interire reliquos tres, in 
quo nullum absurdum sit, cum nee in illorum duratione habeatur saltus, nee in saltu per 
interitum habeatur absurdi quidpiam, ejus exemplo, quod superius dictum fuit, ubi ostensum 
est, non conjungi non esse simul, & esse. Nam in primis 12 gradus velocitatis non sunt quid 
compositum e duodecim rebus inter se distinctis, atque disjunctis, quarum 9 manere possint, 
3 interire, sed sunt unica determinatio ad existendum in punctis spatii distantibus certo 
intervallo, ut palmorumi2, elapsis datis quibusdam temporibus aequalibus quibusvis. Sic 
etiam in ordinatis GD, HE, quae exprimunt velocitates in fig. 6, revera, in mea potissimuim 
Theoria, ordinata GD non est quaedam pars ordinatae HE communis ipsi usque ad D, sed 
sunt duae ordinatae, quarum prima constitit in relatione distantiaa, puncti curvae D a puncto 
axis G, secunda in relatione puncti curvae E a puncto axis H, quod estibi idem, ac punctum G. 


instant of time ; were it not for the fact that that particular propensity is for other reasons 
altered by the conjunction of another cause, which acts at the time, or has already done so ; 
& then another propensity, which is termed compound, will take the place of the former. 
But the absolute propensity, which arises from the combination of all the past & present 
circumstances of the moving body for that instant, is but a single propensity for existing at 
any prescribed instant of subsequent time in a certain prescribed point of space ; & this 
state is absolute for all past & present circumstances, although it may be conditional for 
future circumstances. That is to say, if the same or other causes, acting during subsequent 
instants, do not change that propensity, & the point of space to which it ought to get 
thereafter at the given instant of time, & which it actually does reach if these causes have 
no other effect. Further, it is clear that we cannot have two such absolute states, arising 
from all past & present circumstances, at the same time without prescribing replication ; 
& this the conditional state arising from each of the component velocities does not induce 
because of the very fact that it is conditional. If now there should be a jump from the 
velocity, arising out of all the past & present circumstances, which, after one minute for 
example, compels a point of space to move through 6 palms, to a velocity that compels the 
point to move through 9 palms ; then, at the instant of time, in which the sudden change 
takes place, there would be each of two absolute propensities in respect of all the circum- 
stances of that instant & all that had gone before, existing simultaneously. For in the 
whole of the preceding time there would have been a real series of states having the former 
velocity as a term, & in the whole of the subsequent time there must be one having the 
latter velocity as a term ; hence at that particular instant each of them must occur at one 
& the same time, since neither real series can stand good without each having its own 
real end term. 

66. Again, it is at least possible that the actual velocity of a body, or of an existing At any instant an 
point, may be nothing ; that is to say, if the motion is non-uniform. Now, this always ^^l *? ""** 
is the case in Nature ; as I think can be proved, but it does not concern us at present. But, arising from a kind 
at any rate, it is bound to have some potential velocity, or at least some state, which, y P tentlal vel - 
although usually referred to by another name, & the velocity stated to be nothing, yet is 

not definitely nothing, but is a real state, namely, a propensity for rest. I have come to 
the conclusion, however, that in Nature there is not really such a thing as this state, or 
absolute rest, from arguments that I gave in the Supplements to Stay's Philosophy in 
two paragraphs concerning space & time ; & these I will add at the end of the work, amongst 
some matters, that I will call by the name of supplements in this work as well ; they will 
be placed first & second amongst them. But that idea also does not concern us at present. 
Now, putting on one side these considerations altogether, it follows from the rest of what 
I have said that, if we admit both uniform motion & rest as existing in Nature, or even 
possible, then each velocity must have conditions that necessarily lead to the conclusion 
that according to the argument given above in support of the Law of Continuity it has its 
own corresponding force, & that no passage from one velocity to another can be made 
except through intermediate stages. 

67. Further, it is quite clear that from this it can be rigorously proved that the whole Rigorous proof that 

e i . i . . . , 9 J r , . it is impossible to 

velocity of a body cannot perish or arise in an instant of time, nor for a point that does pas s from one veio- 
not perish or arise along with it ; nor can our arguments with regard to production & cit y to a* 11 ? 1 in 

1-1 i r i T-i i.** an instant of time. 

destruction be made to refer to this. For, since that no velocity of a body, or of an 
existing point, is not absolutely nothing, as I remarked, but is some real state ; & this real 
state is bound to be connected with that other real state, namely, that of the prescribed 
velocity that is being created or destroyed. Hence it comes about that there can be no 
escape from the arguments I have given above, by saying that when the change from twelve 
degrees of velocity is made to nine degrees, the first nine at least endure, whilst the 
remaining three are destroyed ; & then by asserting that there is nothing absurd in this, 
since neither in the duration of the former has there been any sudden change, nor is there 
anything absurd in the jump caused by the destruction of the latter, according to the instance 
of it given above, where it was shown that non-existence & existence must be disconnected. 
For in the first place those twelve degrees of velocity are not something compounded of 
twelve things distinct from, & unconnected with, one another, of which nine can endure 
& three can be destroyed ; but are a single propensity for existing, after the lapse of any 
given number of equal times of any given length, in points of space at a certain interval, 
say twelve palms, away from the original position. So also, with regard to the ordinates 
GD, HE, which in Fig. 6. express velocities, it is the fact that (most especially in my Theory) 
the ordinate GD is not some part of the ordinate HE, common with it as far as the point 
D ; but there are two ordinates, of which the first depends upon the relation of the distance 
of the point D of the curve from the point G on the axis, & the second upon the relation 
of the distance of point E on the curve from the point H on the axis, which is here the 



Relationem distantiae punctorum D, & G constituunt duo reales modi existendi ipsorum, 
relationem distantias punctorum D. & E duo reales modi existendi ipsorum, & relationem 
distantiae punctorum H, & E duo reales modi existendi ipsorum. Haec ultima relatio 
constat duobus modis realibus tantummodo pertinentibus ad puncta E, & H, vel G, & 
summa priorum constat modis realibus omnium trium, E, D, G. Sed nos indefinite con- 
cipimus possibilitatem omnium modorum realium intermediorum, ut infra dicemus, in qua 
praecisiva, & indefinita idea stat mini idea spatii continui ; & intermedii modi possibles inter 
G, & D sunt pars intermediorum inter E, & H. Praeterea omissis etiam hisce omnibus ipse 
ille saltus a velocitate finita ad nullam, vel a nulla ad finitam, haberi non potest. 

Cur adhibita col- 68. Atque hinc ego quidem potuissem etiam adhibere duos globos asquales, qui sibi 

eaiuicm^aKanTpro mv * cem occurrant cum velocitatibus sequalibus, quae nimirum in ipso contactu deberent 

Thcoria deducenda. momento temporis intcrirc ; sed ut hasce ipsas considerationes evitarem de transitu a statu 

reali ad statum itidem realem, ubi a velocitate aliqua transitur ad velocitatem nullam ; 

adhibui potius [32] in omnibus dissertationibus meis globum, qui cum 12 velocitatis gradibus 

assequatur alterum praecedentem cum 6 ; ut nimirum abeundo ad velocitatem aliam 

quamcunque haberetur saltus ab una velocitate ad aliam, in quo evidentius esset absurdum. 

Quo pacto mutata 
velocitate poten- 
tial! per saltum, 
non mutetur per 
saltum actualis. 

69. Jam vero in hisce casibus utique haberi deberet saltus quidam, & violatio legis 
continuitatis, non quidem in velocitate actuali, sed in potentiali, si ad contactum deveniretur 
cum velocitatum discrimine aliquo determinato quocunque. In velocitate actuali, si earn 
metiamur spatio, quod conficitur, diviso per tempus, transitus utique fieret per omnes 
intermedias, quod sic facile ostenditur ope Geometriae. In fig. 10 designent AB, BC bina 
tempora ante & post contactum, & momento quolibet H sit velocitas potentialis ilia major 
HI, quae aequetur velocitati primae AD : quovis autem momento Q posterioris temporis sit 

velocitas potentialis minor QR, quae aequetur 
velocitati cuidam data: CG. Assumpto quovis 
tempore HK determinatae magnitudinis, area 
IHKL divisa per tempus HK, sive recta HI, 
exhibebit velocitatem actualem. Moveatur 
tempus HK versus B, & donee K adveniat ad 
B, semper eadem habebitur velocitatis men- 
sura ; eo autem progressoin O ultra B, sed adhuc 
H existente in M citra B, spatium illi tem- 
pori respondens componetur ex binis MNEB, 
BFPO, quorum summa si dividatur per MO ; 
jam nee erit MN aequalis priori AD, nee BF, 
ipsa minor per datam quantitatem FE ; sed 
facile demonstrari potest (&), capta VE asquali 

D! ~ L V N E Y 

Irrcgularitas alia 
in cxpressione act- 
ualis velocitatis. 






p; R T G 







FIG. 10. 

IL, vel HK, sive MO, & ducta recta VF, quae secet MN in X, quotum ex illo divisione 
prodeuntem fore MX, donee, abeunte toto illo tempore ultra B in QS, jam area QRTS 
divisa per tempus QS exhibeat velocitatem constantem QR. 

70. Patet igitur in ea consideratione a velocitate actuali praecedente HI ad sequentem 
QR transiri per omnes intermedias MX, quas continua recta VF definiet ; quanquam ibi 
etiam irregulare quid oritur inde, quod velocitas actualis XM diversa obvenire debeat pro 
diversa magnitudine temporis assumpti HK, quo nimirum assumpto majore, vel minore 
removetur magis, vel minus V ab E, & decrescit, vel crescit XM. Id tamen accidit in 
motibus omnibus, in quibus velocitas non manet eadem toto tempore, ut nimirum turn 
etiam, si velocitas aliqua actualis debeat agnosci, & determinari spatio diviso per tempus ; 
pro aliis, atque aliis temporibus assumptis pro mensura alias, atque alias velocitatis actualis 
mensuras ob-[33]-veniant, secus ac accidit in motu semper aequabili, quam ipsam ob causam, 
velocitatis actualis in motu difformi nulla est revera mensura accurata, quod supra innui 
sed ejus idea praecisa, ac distincta aequabilitatem motus requirit, & idcirco Mechanic! in 
difformibus motibus ad actualem velocitatem determinandam adhibere solent spatiolum 
infinitesimo tempusculo percursum, in quo ipso motum habent pro aequabili. 

(b) Si enim producatur OP usque ad NE in T, erit ET = VN, ob VE = MO =NT. Est autem 

VE : VN : : EF : NX ; quart VN X EF = VE X NX, sive posito ET pro VN, W MO pro VE, erit 
ET XEF =MO X NX. Totum MNTO est MO X MN, pars FETP est = EY X EF. Quafe residuus 
gnomon NMOPFE est MOx(MN-NX), sive est MO X MX, quo diviso per MO babetur MX. 


same as the point G. The relation of the distance between the points D & G is determined 
by the two real modes of existence peculiar to them, the relation of the distance between 
the points D & E by the two real modes of existence peculiar to them, & the relation of 
the distance between the points H & E by the two real modes of existence peculiar to them. 
The last of these relations depends upon the two real modes of existence that pertain to the 
points E & H (or G), & upon these alone ; the sum of the first & second depends upon all 
three of the modes of the points E, D, & G. But we have some sort of ill-defined conception 
of the possibility of all intermediate real modes of existence, as I will remark later ; & on 
this disconnected & ill-defined idea is founded my conception of continuous space ; also 
the possible intermediate modes between G & D form part of those intermediate between 
E & H. Besides, omitting all considerations of this sort, -that sudden change from a finite 
velocity to none at all, or from none to a finite, cannot happen. 

68. Hence I might just as well have employed two equal balls, colliding with one wh y the collision 
another with equal velocities, which in truth at the moment of contact would have to be the b sameTirecfion 
destroyed in an instant of time. But, in order to avoid the very considerations just stated is employed for the 
with regard to the passage from a real state to another real state (when we pass from a In " 
definite velocity to none), I have preferred to employ in all my dissertations a ball having 

1 2 degrees of velocity, which follows another ball going in front of it with 6 degrees ; 
so that, by passing to some other velocity, there would be a sudden change from one 
velocity to another ; & by this means the absurdity of the idea would be made more 

69. Now, at least in such cases as these, there is bound to be some sudden change & 

a breach of the Law of Continuity, not indeed in the actual velocity, but in the potential sudden change in 
velocity, if the collision occurs with any given difference of velocities whatever. In the ^ ^^T^ might 
actual velocity, measured by the space traversed divided by the time, the change will at any not 'be a sudden 
rate be through all intermediate stages ; & this can easily be shown to be 50 by the aid of ^^veioclty 16 ***" 

In Fig. 10 let AB, BC represent two intervals of time, respectively before & after 
contact ; & at any instant let the potential velocity be the greater velocity HI, equal to the . 
first velocity AD ; & at any instant Q of the time subsequent to contact let the potential 
velocity be the less velocity QR, equal to some given velocity CG. If any prescribed interval 
of time HK be taken, the area IHKL divided by the time HK, i.e., the straight line HI, 
will represent the actual velocity. Let the time HK be moved towards B ; then until 
K comes to B, the measure of the velocity will always be the same. If then, K goes on 
beyond B to O, whilst H still remains on the other side of B at M ; then the space corre- 
sponding to that time will be composed of the two spaces MNEB, BFPO. Now, if the 
sum of these is divided by MO, the result will not be equal to either MN (which is equal 
to the first AD), or BF (which is less than MN by the given quantity FE). But it can 
easily be proved ( ) that, if VE is taken equal to IL, or HK, or MO, & the straight line 
VF is drawn to cut MN in X ; then the quotient obtained by the division will be MX. 
This holds until, when the whole of the interval of time has passed beyond B into the 
position QS, the area QRTS divided by the time QS now represents a constant velocity 
equal to QR. 

70. From the foregoing reasoning it is therefore clear that the change from the A further irregu- 
preceding actual velocity HI to the subsequent velocity QR is made through all intermediate lari ty m the repre- 

r , . . TV/TTT- i i MI i i i i i >> sentation of actual 

velocities such as MX, which will be determined by the continuous straight line VF. There velocity, 
is, however, some irregularity arising from the fact that the actual velocity XM must turn 
out to be different for different magnitudes of the assumed interval of time HK. For, 
according as this is taken to be greater or less, so the point V is removed to a greater or 
less distance from E ; & thereby XM will be decreased or increased correspondingly. This 
is the case, however, for all motions in which the velocity does not remain the same during 
the whole interval ; as for instance in the case where, if any actual velocity has to be found 
& determined by the quotient of the space traversed divided by the time taken, far other 
& different measures of the actual velocities will arise to correspond with the different 
intervals of time assumed for their measurement ; which is not the case for motions that 
are always uniform. For this reason there is no really accurate measure of the actual 
velocity in non-uniform motion, as I remarked above ; but a precise & distinct idea of it 
requires uniformity of motion. Therefore Mechanicians in non-uniform motions, as a 
means to the determination of actual velocity, usually employ the small space traversed in 
an infinitesimal interval of time, & for this interval they consider that the motion is uniform. 

(b) For if OP be produced to meet NE in T, then EY = VN ; for VE = MO = NT. Moreover 

VE : VN=EF : NX ; and therefore VN.EF=VE.NX. Hence, replacing VN hy EY, and. VE hy MO, we have 
EYEF=MO.NX. Now, the whole MNYO = MO.MN, and the part FEYP= ET.EF. Hence the remainder 
(the gnomon NMOPFE) = MO.(MN NX) = MO.MX .- and this, on division by MO, will give MX. 


" \mmc yi- At velocitas potcntialis, quas singulis momentis temporis respondet sua, mutaretur 

citatum non posse utique per saltum ipso momento B, quo deberet haberi & ultima velocitatum praecedentium 
entia ni vciodtatum r " ^' ^ P" ma sequentium BF, quod cum haberi nequeat, uti demonstratum est, fieri non 
potest per secundum ex argumentis, quae adhibuimus pro lege continuitatis, ut cum ilia 
velocitatum inasqualitate deveniatur ad immediatum contactum ; atque id ipsum excludit 
etiam inductio, quam pro lege continuitatis in ipsis quoque velocitatibus, atque motibus 
primo loco proposui. 

Prpmovenda ana- 72. Atque hoc demum pacto illud constitit evidenter, non licere continuitatis legem 

deserere in collisione corporum, & illud admittere, ut ad contactum immediatum deveniatur 
cum illaesis binorum corporum velocitatibus integris. Videndum igitur, quid necessario 
consequi debeat, ubi id non admittatur, & haec analysis ulterius promovenda. 

ifaberimu- 73' Q uon i am a ^ immediatum contactum devenire ea corpora non possunt cum praece- 

tationem veiocita- dentibus velocitatibus ; oportet, ante contactum ipsum immediatum incipiant mutari 
auk mutat Ue Vlm> velocitates ipsae, & vel ea consequentis corporis minui, vel ea antecedentis augeri, vel 
utrumque simul. Quidquid accidat, habebitur ibi aliqua mutatio status, vel in altero 
corpore, vel in utroque, in ordine ad motum, vel quietem, adeoque habebitur aliqua 
mutationis causa, quaecunque ilia sit. Causa vero mutans statum corporis in ordine ad 
motum, vel quietem, dicitur vis ; habebitur igitur vis aliqua, quae effectum gignat, etiam 
ubi ilia duo corpora nondum ad contactum devenerint. 

Earn vim debere 74. Ad impediendam violationem continuitatis satis esset, si ejusmodi vis ageret in 

. iSf-SSi & alterum tantummodo e binis corporibus, reducendo praecedentis velocitatem ad gradus 12, 

agere m panes op- . r. . ' . 

positas. vel sequentis ad 6. Videndum igitur aliunde, an agere debeat in alterum tantummodo, an 

in utrumque simul, & quomodo. Id determinabitur per aliam Naturae legem, quam nobis 
inductio satis ampla ostendit, qua nimirum evincitur, omnes vires nobis cognitas agere 
utrinque & aequaliter, & in partes oppositas, unde provenit principium, quod appellant 
actionis, & reactionis aequalium ; est autem fortasse quaedam actio duplex semper aequaliter 
agens in partes oppositas. Ferrum, & magnes aeque se mutuo trahunt ; elastrum binis 
globis asqualibus interjectum aeque utrumque urget, & aequalibus velocitatibus propellit ; 
gravitatem ipsam generalem mutuam esse osten-[34]-dunt errores Jovis, ac Saturni potissi- 
mum, ubi ad se invicem accedunt, uti & curvatura orbitae lunaris orta ex ejus gravitate in 
terram comparata cum aestu maris orto ex inaequali partium globi terraquei gravitate in 
Lunam. Ipsas nostrae vires, quas nervorum ope exerimus, semper in partes oppositas agunt, 
nee satis valide aliquid propellimus, nisi pede humum, vel etiam, ut efficacius agamus, 
oppositum parietem simul repellamus. En igitur inductionem, quam utique ampliorem 
etiam habere possumus, ex qua illud pro eo quoque casu debemus inferre, earn ibi vim in 
utrumque corpus agere, quae actio ad aequalitatem non reducet inaequales illas velocitates, 
nisi augeat praecedentis, minuat consequentis corporis velocitatem ; nimirum nisi in iis 
producat velocitates quasdam contrarias, quibus, si solae essent, deberent a se invicem 
recedere : sed quia eae componuntur cum praecedentibus ; hasc utique non recedunt, sed 
tantummodo minus ad se invicem accedunt, quam accederent. 

Hinc dicendam 75. Invenimus igitur vim ibi debere esse mutuam, quae ad partes oppositas agat, & quae 


sua na t ura determinet per sese ilia corpora ad recessum mutuum a se invicem. Hujusmodi 

quaerendam ejus . . . . . ./ . . 11 i /~ j i 

legem. igitur vis ex nomims denmtione appellari potest vis repulsiva. Uuaerendum jam ulterius, 

qua lege progredi debeat, an imminutis in immensum distantiis ad datam quandam mensuram 
deveniat, an in infinitum excrescat ? 

Ea vi debere totum 76. Ut in illo casu evitetur saltus ; satis est in allato exemplo ; si vis repulsiva, ad quam 

crimen at eHdi ante delati sumus, extinguat velocitatum differentiam illam 6 graduum, antequam ad contactum 

contactum. immediatum corpora devenirent : quamobrem possent utique devenire ad eum contactum 

eodem illo momento, quo ad aequalitatem velocitatum deveniunt. At si in alio quopiam 

casu corpus sequens impellatur cum velocitatis gradibus 20, corpore praecedente cum suis 6 ; 


71. The potential velocity, each corresponding to its own separate instant of time, The conclusion is 

ij f i j jj i ^.t i. t n a i t nat immediate 

would certainly be changed suddenly at that instant ot time r> ; & at this point we are contact with a dif- 

bound to have both the last of the preceding velocities, BE, & the first of the subsequent ference of velocities 

velocities, BF. Now, since (as has been already proved) this is impossible, it follows from 

the second of the arguments that I used to prove the Law of Continuity, that it cannot 

come about that the bodies come into immediate contact with the inequality of velocities 

in question. This is also excluded by induction, such as I gave in the first place for the 

Law of Continuity, in the case also of these velocities & motions. 

72. In this manner it is at length clearly established that it is not right to neglect the immediate contact 
Law of Continuity in the collision of bodies, & admit the idea that they can come into ^Sysis^tobe ca^ 
immediate contact with the whole velocities of both bodies unaltered. Hence, we must ried further, 
now investigate the consequences that necessarily follow when this idea is not admitted ; 

& the analysis must be carried further. 

73. Since the bodies cannot come into immediate contact with the velocities they had There must be then, 
at first, it is necessary that those velocities should commence to change before that immediate change in the v'eio a 
contact ; & either that of the body that follows should be diminished, or that of the one cit y ' & therefore 
going in front should be increased, or that both these changes should take place together, causes the change! 1 
Whatever happens, there will be some change of state at the time, in one or other of the 

bodies, or in both, with regard to motion or rest ; & so there must be some cause for this 
change, whatever it is. But a cause that changes the state of a body as regards motion or 
rest is called force. Hence there must be some force, which gives the effect, & that too 
whilst the two bodies have not as yet come into contact. 

74. It would be enough, to avoid a breach of the Law of Continuity, if a force of The f rce o must V 6 

i i.ii 11 r -I IT i i i i < i i i mutual, & act m 

this kind should act on one of the two bodies only, altering the velocity of the body in opposite directions, 
front to 12 degrees, or that of the one behind to 6 degrees. Hence we must find out, 
from other considerations, whether it should act on one of the two bodies only, or on both 
of them at the same time, & how. This point will be settled by another law of Nature, 
which sufficiently copious induction brings before us ; that is, the law in which it is estab- 
lished that all forces that are known to us act on both bodies, equally, and in opposite 
directions. From this comes the principle that is called ' the principle of equal action 
& reaction ' ; perchance this may be a sort of twofold action that always produces its 
effect equally in opposite directions. Iron & a loadstone attract one another with the 
same strength ; a spring introduced between two balls exerts an equal action on either 
ball, & generates equal velocities in them. That universal gravity itself is mutual is proved 
by the aberrations of Jupiter & of Saturn especially (not to mention anything else) ; that 
is to say, the way in which they err from their orbits & approach one another mutually. 
So also, when the curvature of the lunar orbit arising from its gravitation towards the 
Earth is compared with the flow of the tides caused by the unequal gravitation towards 
the Moon of different parts of the land & water that make up the Earth. Our own bodily 
forces, which produce their effect by the help of our muscles, always act in opposite direc- 
tions ; nor have we any power to set anything in motion, unless at the same time we press 
upon the earth with our feet or, in order to get a better purchase, upon something that 
will resist them, such as a wall opposite. Here then we have an induction, that can be 
made indeed more ample still ; & from it we are bound in this case also to infer that the 
force acts on each of the two bodies. This action will not reduce to equality those two 
unequal velocities, unless it increases that of the body which is in front & diminishes that 
of the one which follows. That is to say, unless it produces in them velocities that are 
opposite in direction ; & with these velocities, if they alone existed, the bodies would 
move away from one another. But, as they are compounded with those they had to start 
with, the bodies do not indeed recede from one another, but only approach one another 
less quickly than they otherwise would have done. 

75. We have then found that the force must be a mutual force which acts in opposite Hence the force 
directions ; one which from its very nature imparts to those bodies a natural propensity p u * sive *r ^"1^ 
for mutual recession from one another. Hence a force of this kind, from the very meaning governing it is now 
of the term, may be called a repulsive force. We have now to go further & find the law to ^ found - 
that it follows, & whether, when the distances are indefinitely diminished, it attains any 

given measure, or whether it increases indefinitely. 

76. In this case, in order that any sudden change may be avoided, it is sufficient, in The whole differ- 
the example under consideration, if the repulsive force, to which our arguments have led veiocities W must *be 
us, should destroy that difference of 6 degrees in the velocities before the bodies should destroyed by the 
have come into immediate contact. Hence they might possibly at least come into contact t ^ e 

at the instant in which they attained equality between the velocities. But if in another 
case, say, the body that was behind were moving with 20 degrees of velocity, whilst the 


turn vero ad contactum deveniretur cum differentia velocitatum majore, quam graduum 8. 
Nam illud itidem amplissima inductione evincitur, vires omnes nobis cognitas, quas aliquo 
tempore agunt, ut velocitatem producant, agere in ratione temporis, quo agunt, & sui 
ipsius. Rem in gravibus oblique descendentibus experimenta confirmant ; eadem & in 
elastris institui facile possunt, ut rem comprobent ; ac id ipsum est fundamentum totius 
Mechanicae, quae inde motuum leges eruit, quas experimenta in pendulis, in projectis 
gravibus, in aliis pluribus comprobant, & Astronomia confirmat in caelestibus motibus. 
Quamobrem ilia vis repulsiva, quae in priore casu extinxit 6 tantummodo gradus discriminis, 
si agat breviore tempore in secundo casu, non poterit extinguere nisi pauciores, minore 
nimirum velocitate producta utrinque ad partes contrarias. At breviore utique tempore 
aget : nam cum majore velocitatum discrimine velocitas respectiva est major, ac proinde 
accessus celerior. [35] Extingueret igitur in secundo casu ilia vis minus, quam 6 discriminis 
gradus, si in primo usque ad contactum extinxit tantummodo 6. Superessent igitur plures, 
quam 8 ; nam inter 20 & 6 erant 14, ubi ad ipsum deveniretur contactum, & ibi per saltum 
deberent velocitates mutari, ne compenetratio haberetur, ac proinde lex continuitatis 
violari. Cum igitur id accidere non possit ; oportet, Natura incommodo caverit per 
ejusmodi vim, quae in priore casu aliquanto ante contactum extinxerit velocitatis discrimen, 
ut nimirum imminutis in secundo casu adhuc magis distantiis, vis ulterior illud omne 
discrimen auferat, elisis omnibus illis 14 gradibus discriminis, qui habebantur. 

Earn vim debere 
augeri in infinitum, 
imminutis, & qui- 
dem in infinitum, 
distantiis : habente 
virium curva ali- 
quam asymptotum 
in origine abscissa- 

77. Quando autem hue jam delati sumus, facile est ulterius progredi, & illud con- 
siderare, quod in secundo casu accidit respectu primi, idem accidere aucta semper velocitate 
consequentis corporis in tertio aliquo respectu secundi, & ita porro. Debebit igitur ad 
omnem pro omni casu evitandum saltum Natura cavisse per ejusmodi vim, quae imminutis 
distantiis crescat in infinitum, atque ita crescat, ut par sit extinguendas cuicunque velocitati, 
utcunque magnae. Devenimus igitur ad vires repulsivas imminutis distantiis crescentes 
in infinitum, nimirum ad arcum ilium asymptoticum ED curae virium in fig. i propositum. 
Illud quidem ratiocinatione hactenus instituta immediate non deducitur, hujusmodi 
incrementa virium auctarum in infinitum respondere distantiis in infinitum imminutis. 
Posset pro hisce corporibus, quae habemus prae manibus, quasdam data distantia quascunque 
esse ultimus limes virium in infinitum excrescentium, quo casu asymptotus AB non transiret 
per initium distantiae binorum corporum, sed tanto intervallo post ipsum, quantus esset 
ille omnium distantiarum, quas remotiores particulse possint acquirere a se invicem, limes 
minimus ; sed aliquem demum esse debere extremum etiam asymptoticum arcum curvas 
habentem pro asymptote rectam transeuntem per ipsum initium distantiae, sic evincitur ; 
si nullus ejusmodi haberetur arcus ; particulae materiae minores, & primo collocatae in 
distantia minore, quam esset ille ultimus limes, sive ilia distantia asymptoti ab initio 
distantias binorum punctorum materiae, in mutuis incursibus velocitatem deberent posse 
mutare per saltum, quod cum fieri nequeat, debet utique aliquis esse ultimus asymptoticus 
arcus, qui asymptotum habeat transeuntem per distantiarum initium, & vires inducat 
imminutis in infinitum distantiis crescentes in infinitum ita, ut sint pares velocitati extin- 
guendae cuivis, utcunque magnae. Ad summum in curva virium haberi possent plures 
asymptotici arcus, alii post alios, habentes ad exigua intervalla asymptotes inter se parallelas, 
qui casus itidem uberrimum aperit contemplationibus fcecundissimis campum, de quo 
aliquid inferius ; sed aliquis arcus asympto-[36]-ticus postremus, cujusmodi est is, quern 
in figura i proposui, haberi omnino debet. Verum ea perquisitione hie omissa, pergendum 
est in consideratione legis virium, & curvae earn exprimentis, quae habentur auctis distantiis. 

vim in majoribus 
tractfvam, ^ 

78. In primis gravitas omnium corporum in Terram, quam quotidie experimur, satis 
, evmc i t > repulsionem illam, quam pro minimis distantiis invenimus, non extendi ad distantias 

secante axem in quascunque, sed in magnis jam distantiis haberi determinationem ad accessum, quam vim 
aliquo hmite. attractivam nominavimus. Quin immo Keplerianae leges in Astronomia tarn feliciter a 

Newtono adhibitae ad legem gravitatis generalis deducendam, & ad cometas etiam traductas, 









body in front still had its' original 6 degrees ; then they would come into contact with 
a difference of velocity greater than 8 degrees. For, it can also be proved by the fullest 
possible induction that all forces known to us, which act for any intervals of time so as to 
produce velocity, give effects that are proportional to the times for which they act, & also 
to the magnitudes of the forces themselves. This is confirmed by experiments with heavy 
bodies descending obliquely ; the same things can be easily established in the case of springs 
so as to afford corroboration. Moreover it is the fundamental theorem of the whole of 
Mechanics, & from it are derived the laws of motion ; these are confirmed by experiments 
with pendulums, projected weights, & many other things ; they are corroborated also by 
astronomy in the matter of the motions of the heavenly bodies. Hence the repulsive force, 
which in the first case destroyed only 6 degrees difference of velocity, if it acts for a shorter 
time in the second case, will not be able to destroy aught but a less number of degrees, as 
the velocity produced in the two bodies in opposite directions is less. Now it certainly 
will act for a shorter time ; for, owing to the greater difference of velocities, the relative 
velocity is greater & therefore the approach is faster. Hence, in the second case the force 
would destroy less than 6 degrees of the difference, if in the first case it had, just at contact, 
destroyed 6 degrees only. There would therefore be more than 8 degrees left over (for, 
between 20 & 6 there are 14) when contact happened, & then the velocities would have 
to be changed suddenly unless there was compenetration ; & thereby the Law of Continuity 
would be violated. Since, then, this cannot be the case, Nature would be sure to guard 
against this trouble by a force of such a kind as that which, in the former case, extinguished 
the difference of velocity some time before contact ; that is to say, so that, when the 
distances are still further diminished in the second case, a further force eliminates all 
that difference, all of the 14 degrees of difference that there were originally being 

77. Now, after that we have been led so far, it is easy to go on further still & to consider ' nie fon : e mus * "* 
that, what happens in the second case when compared with the first, will happen also in SThe distances Ire 
a third case, in which the velocity of the body that follows is once more increased, when diminished, also 
compared with the second case ; & so on, & so on. Hence, in order to guard against any Sn-ve"^* 6 forces has 

sudden change at all in every case whatever, Nature will necessarily have taken measures an asymptote at the 

for this purpose by means of a force of such a kind that, as the distances are diminished the ongm 

force increases indefinitely, & in such a manner that it is capable of destroying any velocity, 

however great it may be. We have arrived therefore at repulsive forces that increase as 

the distances diminish, & increase indefinitely ; that is to say, to the asymptotic arc, ED, 

of the curve of forces exhibited in Fig. i . It is indeed true that by the reasoning given so 

far it is not immediately deduced that increments of the forces when increased to infinity 

correspond with the distances diminished to infinity. There may be for these bodies, 

such as we have in consideration, some fixed distance that acts as a boundary limit to forces 

that increase indefinitely ; in this case the asymptote AB will not pass through the 

beginning of the distance between the two bodies, but at an interval after it as great as the 

least limit of all distances that particles, originally more remote, might acquire from one 

another. But, that there is some final asymptotic arc of the curve having for its asymptote 

the straight line passing through the very beginning of the distance, is proved as follows. 

If there were no arc of this kind, then the smaller particles of matter, originally collected 

at a distance less than this final limit would be, i.e., less than the distance of the asymptote 

from the beginning of the distance between the two points of matter, must be capable of 

having- their velocities, on collision with one another, suddenly changed. Now, as this is 

impossible, then at any rate there must be some asymptotic arc, which has an asymptote 

passing through the very beginning of the distances ; & this leads us to forces that, as the 

distances are indefinitely diminished, increase indefinitely in such a way that they are 

capable of destroying any velocity, no matter how large it may be. In general, in a curve 

of forces there may be several asymptotic arcs, one after the other, having at short intervals 

asymptotes parallel to one another ; & this case also opens up a very rich field for fruitful 

investigations, about which I will say something later. But there must certainly be some 

one final asymptotic arc of the kind that I have given in Fig. i. However, putting 

this investigation on one side, we must get on with the consideration of the law 

of forces, & the curve that represents them, which are obtained when the distances 

are increased. 

78. First of all, the gravitation of all bodies towards the Earth, which is an everyday The force at greater 
experience, proves sufficiently that the repulsion that we found for very small distances fv^he^curve^cut- 
does not extend to all distances ; but that at distances that are now great there is a ting the axis at 
propensity for approach, which we have called an attractive force. Moreover the Keplerian s 
Laws in astronomy, so skilfully employed by Newton to deduce the law of universal 
gravitation, & applied even to the comets, show perfectly well that gravitation extends, 


satis ostendunt, gravitatem vel in infinitum, vel saltern per totum planetarium, & come- 
tarium systema extendi in ratione reciproca duplicata distantiarum. Quamobrem virium 
curva arcum habet aliquem jacentem ad partes axis oppositas, qui accedat, quantum sensu 
percipi possit, ad earn tertii gradus hyperbolam, cujus ordinatae sunt in ratione reciproca 
duplicata distantiarum, qui nimirum est ille arcus STV figuras I. Ac illud etiam hinc 
patet, esse aliquem locum E, in quo curva ejusmodi axem secet, qui sit limes attractionum, 
& repulsionum, in quo ab una ad alteram ex iis viribus transitus fiat. 

Plures esse debere, 79. Duos alios nobis indicat limites ejusmodi, sive alias duas intersectiones, ut G & I, 

linStes 3 P n3enomen um vaporum, qui oriuntur ex aqua, & aeris, qui a fixis corporibus gignitur ; 
cum in iis ante nulla particularum repulsio fuerit, quin immo fuerit attractio, ob 
cohaerentiam, qua, una parte retracta, altera ipsam consequebatur, & in ilia tanta expansione, 
& elasticitatis vi satis se manifesto prodat repulsio, ut idcirco a repulsione in minimis distantiis 
ad attractionem alicubi sit itum, turn inde iterum ad repulsionem, & iterum inde ad generalis 
gravitatis attractiones. Effervescentiae, & fermentationes adeo diversae, in quibus cum 
adeo diversis velocitatibus eunt, ac redeunt, & jam ad se invicem accedunt, jam recedunt 
a se invicem particulae, indicant utique ejusmodi limites, atque transitus multo plures ; 
sed illos prorsus evincunt substantise molles, ut cera, in quibus compressiones plurimse 
acquiruntur cum distantiis admodum adversis, in quibus, tamen omnibus limites haberi 
debent ; nam, anteriore parte ad se attracta, posteriores earn sequuntur, eadem propulsa, 
illae recedunt, distantiis ad sensum non mutatis, quod ob illas repulsiones in minimis 
distantiis, quae contiguitatem impediunt, fieri alio modo non potest, nisi si limites ibidem 
habeantur in iis omnibus distantiis inter attractiones, & repulsiones, quae nimirum requi- 
runtur ad hoc, ut pars altera alteram consequatur retractam, vel prsecedat propulsam. 

Hinc tota curvae 80. Habentur igitur plurimi limites, & plurimi flexus curvse hinc, & inde ab axe prseter 

ayroptotL m & b Tu- ^ uos arcus > quorum prior ED in infinitum protenditur, & asymptoticus est, alter STV, 
ribus flexibus, ac [37] si gravitas generalis in infinitum protenditur, est asymptoticus itidem, & ita accedit 
ad crus illud hyperbolae gradus tertii, ut discrimen sensu percipi nequeat : nam cum ipso 
penitus congruere omnino non potest ; non enim posset ab eodem deinde discedere, cum 
duarum curvarum, quarum diversa natura est, nulli arcus continui, utcunque exigui, possint 
penitus congruere, sed se tantummodo secare, contingere, osculari possint in punctis 
quotcunque, & ad se invicem accedere utcumque. Hinc habetur jam tota forma curvae 
virium, qualem initio proposui, directa ratiocinatione a Naturae phsenomenis, & genuinis 
principiis deducta. Remanet jam determinanda constitutio primorum elementorum 
materiae ab iis viribus deducta, quo facto omnis ilia Theoria, quam initio proposui, patebit, 
nee erit arbitraria quaedam hypothesis, ac licebit progredi ad amovendas apparentes quasdam 
difHcultates, & ad uberrimam applicationem ad omnem late Physicam qua exponendam, 
qua tantummodo, ne hoc opus plus aequo excrescat, indicandam. 

Hinc elementorum 81. Quoniam, imminutis in infinitum distantiis, vis repulsiva augetur in infinitum ; 

m facile patet, nullam partem materias posse esse contiguam alteri parti : vis enim ilia repulsiva 


partibus. protinus alteram ab altera removeret. Quamobrem necessario inde consequitur, prima 

materiae elementa esse omnino simplicia, & a nullis contiguis partibus composita. Id 
quidem immediate, & necessario fluit ex ilia constitutione virium, quae in minimis distantiis 
sunt repulsivae, & in infinitum excrescunt. 

Soiutio objectionis 82. Objicit hie fortasse quispiam illud, fieri posse, ut particulae primigenias materias 

petitaeex eo quod s j nt com p O sitae quidem, sed nulla Naturae vi divisibiles a se invicem, quarum altera tota 

vires repulsivas r . . ^ . .... . . . ,. .. i i 

habere possent non respectu altenus totius habeat vires illas in minimis distantiis repulsivas, vel quarum pars 
puncta smguia, se q u3ev i s respectu reliquarum partium eiusdem particulae non solum nullam habeat repulsivam 

particulae primi- T-. 1,1 '-.,, J . r ,. ,.,. r . . 

geniae. vim, sed habeat maximam illam attractivam, qua; ad ejusmodi cohaesionem requintur : 

eo pacto evitari debere quemvis immediatum impulsum, adeoque omnem saltum, & con- 
tinuitatis laesionem. At in primis id esset contra homogeneitatem materiae, de qua agemus 
infra : nam eadem materiae pars in iisdem distantiis respectu quarundam paucissimarum 
partium, cum quibus particulam suam componit, haberet vim repulsivam, respectu autem 


either to infinity or at least to the limits of the system including all the planets & comets, 
in the inverse ratio of the squares of the distances. Hence the curve will have an arc 
lying on the opposite side of the axis, which, as far as can be perceived by our senses, 
approximates to that hyperbola of the third degree, of which the ordinates are in the inverse 
ratio of the squares of the distances ; & this indeed is the arc STV in Fig. i. Now from 
this it is evident that there is some point E, in which a curve of this kind cuts the axis ; 
and this is a limit-point for attractions and repulsions, at which the passage from one to 
the other of these forces is made. 

79. The phenomenon of vapour arising from water, & that of gas produced from There are bound to 
fixed bodies lead us to admit two more of these limit-points, i.e., two other intersections, ^Syof'tiSep^ 
say, at G & I. Since in these there would be initially no repulsion, nay rather there sages, with corre- 
would be an attraction due to cohesion, by which, when one part is retracted, another 1 " 6 hmit 
generally followed it : & since in the former, repulsion is clearly evidenced by the 

greatness of the expansion, & by the force of its elasticity ; it therefore follows that 
there is, somewhere or other, a passage from repulsion at very small distances to attraction, 
then back again to repulsion, & from that back once more to the attractions of universal 
gravitation. Effervescences & fermentations of many different kinds, in which the 
particles go & return with as many different velocities, & now approach towards & 
now recede from one another, certainly indicate many more of these limit-points & 
transitions. But the existence of these limit-points is perfectly proved by the case of 
soft substances like wax ; for in these substances a large number of compressions are acquired 
with very different distances, yet in all of these there must be limit-points. For, if the 
front part is drawn out, the part behind will follow ; or if the former is pushed inwards, 
the latter will recede from it, the distances remaining approximately unchanged. This, on 
account of the repulsions existing at very small distances, which prevent contiguity, can- 
not take place in any way, unless there are limit-points there in all those distances between 
attractions & repulsions ; namely, those that are requisite to account for the fact that one 
part will follow the other when the latter is drawn out, & will recede in front of the 
latter when that is pushed in. 

80. Therefore there are a large number of limit-points, & a large number of flexures Hence we get the 
on the curve, first on one side & then on the other side of the axis, in addition to two whole for h of t ^ e 
arcs, one of which, ED, is continued to infinity & is asymptotic, & the other, STV, is asymptotes, many 
asymptotic also, provided that universal gravitation extends to infinity. It approximates flexures & many 

J i r j- r i i r i i i i -11 111 i intersections with 

to the form of the hyperbola of the third degree mentioned above so closely that the the axis, 
difference from it is imperceptible ; but it cannot altogether coincide with it, because, in 
that case it would never depart from it. For, of two curves of different nature, there 
cannot be any continuous arcs, no matter how short, that absolutely coincide ; they can 
only cut, or touch, or osculate one another in an indefinitely great number of points, & 
approximate to one another indefinitely closely. Thus we now have the whole form of 
the curve of forces, of the nature that I gave at the commencement, derived by a straight- 
forward chain of reasoning from natural phenomena, & sound principles. It only remains 
for us now to determine the constitution of the primary elements of matter, derived from 
these forces ; : in this manner the whole of the Theory that I enunciated at the start 
will become quite clear, & it will not appear to be a mere arbitrary hypothesis. We 
can proceed to remove certain apparent difficulties, & to apply it with great profit to 
the whole of Physics in general, explaining some things fully &, to prevent the work 
from growing to an unreasonable size, merely mentioning others. 

81. Now, because the repulsive force is indefinitely increased when the distances are The simplicity of 
indefinitely diminished, it is quite easy to see clearly that no part of matter can be contiguous ments^oT^att^r " 
to any other part ; for the repulsive force would at once separate one from the other, they are altogether 
Therefore it necessarily follows that the primary elements of matter are perfectly simple, w^ 110 "* P ar ts. 

& that they are not composed of any parts contiguous to one another. This is an 
immediate & necessary deduction from the constitution of the forces, which are repulsive 
at very small distances & increase indefinitely. 

82. Perhaps someone will here raise the objection that it may be that the primary Solution of the ob- 
particles of matter are composite, but that they cannot be disintegrated by any force in j n e tlo ^ SS ertion d that 
Nature; that one whole with regard to another whole may possibly have those forces single points can- 
that are repulsive at very small distances, whilst any one part with regard to any other part ?OTces, a u7 P t h'at 
of the same particle may not only have no repulsive force, but indeed may have a very primary particles 
great attractive force such as is required for cohesion of this sort ; that, in this way, we can have them - 
are bound to avoid all immediate impulse, & so any sudden change or breach of continuity. 

But, in the first place, this would be in opposition to the homogeneity of matter, which 
we will consider later ; for the same part of matter, at the same distances with regard to 
those very few parts, along with which it makes up the particle, would have a repulsive 

8 4 


aliarum omnium attractivam in iisdem distantiis, quod analogic adversatur. Deinde si a 
Deo agente supra vires Naturae sejungerentur illas partes a se invicem, turn ipsius Naturae 
vi in se invicem incurrerent ; haberetur in earum collisione saltus naturalis, utut praesup- 
ponens aliquid factum vi agente supra Naturam. Demum duo turn cohaesionum genera 
deberent haberi in Natura admodum diversa, alterum per attractionem in minimis distantiis, 
alterum vero longe alio pacto in elementarium particularum massis, nimirum per limites 
cohaesionis ; adeoque multo minus simplex, & minus uniformis evaderet Theoria. 

An elementa sint [38] 
extensa : argumen- 
ta pro virtual! eor- 
um extensione. 

83. Simplicitate & incompositione elementorum defmita, dubitari potest, an ea 
sint etiam inextensa, an aliquam, utut simplicia, extensionem habeant ejus generis, quam 
virtualem extensionem appellant Scholastici. Fuerunt enim potissimum inter Peripateticos, 
qui admiserint elementa simplicia, & carentia partibus, atque ex ipsa natura sua prorsus 
indivisibilia, sed tamen extensa per spatium divisibile ita, ut alia aliis ma jus etiam occupent 
spatium, ac eo loco, quo unum stet, possint, eo remote, stare simul duo, vel etiam plura ; 
ac sunt etiamnum, qui ita sentiant. Sic etiam animam rationalem hominis utique prorsus 
indivisibilem censuerunt alii per totum corpus diffusam : alii minori quidem corporis parti, 
sed utique parti divisibili cuipiam, & extensae, praesentem toti etiamnum arbitrantur. 
Deum autem ipsum praesentem ubique credimus per totum utique divisibile spatium, 
quod omnia corpora occupant, licet ipse simplicissimus sit, nee ullam prorsus compositionem 
admittat. Videtur autem sententia eadem inniti cuidam etiam analogiae loci, ac temporis. 
Ut enim quies est conjunctio ejusdem puncti loci cum serie continua omnium moment- 
orum ejus temporis, quo quies durat : sic etiam ilia virtualis extensio est conjunctio unius 
momenti temporis cum serie continua omnium punctorum spatii, per quod simplex illud 
ens virtualiter extenditur ; ut idcirco sicut ilia quies haberi creditur in Natura, ita & haec 
virtualis extensio debeat admitti, qua admissa poterunt utique ilia primse materiae elementa 
esse simplicia, & tamen non penitus inextensa. 

Exciuditur virtu- 

rite appiicato. 

84. At ego quidem arbitror, hanc itidem sententiam everti penitus eodem inductionis 
principio, ex quo alia tarn multa hucusque, quibus usi sumus, deduximus. Videmus enim 
in his corporibus omnibus, quae observare possumus, quidquid distinctum occupat locum, 
distinctum esse itidem ita, ut etiam satis magnis viribus adhibitis separari possint, quae 
diversas occupant spatii partes, nee ullum casum deprehendimus, in quo magna haec corpora 
partem aliquam habeant, quae eodem tempore diversas spatii partes occupet, & eadem 
sit. Porro haec proprietas ex natura sua ejus generis est, ut aeque cadere possit in 
magnitudines, quas per sensum deprehendimus, ac in magnitudines, quae infra sensuum 
nostrorum limites sunt ; res nimirum pendet tantummodo a magnitudine spatii, per quod 
haberetur virtualis extensio, quae magnitudo si esset satis ampla, sub sensus caderet. Cum 
igitur nunquam id comperiamus in magnitudinibus sub sensum cadentibus, immo in 
casibus innumeris deprehendamus oppositum : debet utique res transferri ex inductionis 
principio supra exposito ad minimas etiam quasque materiae particulas, ut ne illae quidem 
ejusmodi habeant virtualem extensionem. 

Responsioadexem- [39] 85. Exempla, quae adduntur, petita ab anima rational}, & ab omnipraesentia 
plum anima & Dei. j) e j } n j^ positive evincunt, cum ex alio entium genere petita sint ; praeterquam quod nee 
illud demonstrari posse censeo, animam rationalem non esse unico tantummodo, simplici, 
& inextenso corporis puncto ita praesentem, ut eundem locum obtineat, exerendo inde 
vires quasdam in reliqua corporis puncta rite disposita, in quibus viribus partim necessariis, 
& partim liberis, stet ipsum animae commercium cum corpore. Dei autem praesentia 
cujusmodi sit, ignoramus omnino ; quem sane extensum per spatium divisibile nequaquam 
dicimus, nee ab iis modis omnem excedentibus humanum captum, quibus ille existit, 
cogitat, vult, agit, ad humanos, ad materiales existendi, agendique modos, ulla esse potest 
analogia, & deductio. 

itidem ad analo- 86. Quod autem pertinet ad analogiam cum quiete, sunt sane satis valida argumenta, 

giam cum quiete. q u ibus, ut supra innui, ego censeam, in Natura quietem nullam existere. Ipsam nee posse 


force ; but it would have an attractive force with regard to all others, at the very same 
distances ; & this is in opposition to analogy. Secondly, if, due to the action of GOD 
surpassing the forces of Nature, those parts are separated from one another, then urged 
by the forces of Nature they would rush towards one another ; & we should have, from 
their collision, a sudden change appertaining to Nature, although conveying a presumption 
that something was done by the action of a supernatural force. Lastly, with this idea, 
there would have to be two kinds of cohesion in Nature that were altogether different in 
constitution ; one due to attraction at very small distances, & the other coming about 
in a far different way in the case of masses of elementary particles, that is to say, due to 
the limit-points of cohesion. Thus a theory would result that is far less simple & less 
uniform than mine. 

83. Taking it for granted, then, that the elements are simple & non-composite, whether the ele- 
there can be no doubt as to whether they are also non-extended or whether, although ments are extended; 

, , , , .' , 1-1 i V certain arguments 

simple, they have an extension of the kind that is termed virtual extension by the m favour of virtual 

Scholastics. For there were some, especially among the Peripatetics, who admitted elements extension. 

that were simple, lacking in all parts, & from their very nature perfectly indivisible ; 

but, for all that, so extended through divisible space that some occupied more room than 

others ; & such that in the position once occupied by one of them, if that one were 

removed, two or even more others might be placed at the same time ; & even now there 

are some who are of the same opinion. So also some thought that the rational soul in 

man, which certainly is altogether indivisible, was diffused throughout the whole of the 

body ; whilst others still consider that it is present throughout the whole of, indeed, a 

smaller part of the body, but yet a part that is at any rate divisible & extended. 

Further we believe that GOD Himself is present everywhere throughout the whole of the 

undoubtedly divisible space that all bodies occupy ; & yet He is onefold in the highest 

degree & admits not of any composite nature whatever. Moreover, the same idea seems 

to depend on an analogy between space & time. For, just as rest is a conjunction with 

a continuous series of all the instants in the interval of time during which the rest endures ; 

so also this virtual extension is a conjunction of one instant of time with a continuous series 

of all the points of space throughout which this one-fold entity extends virtually. Hence, 

just as rest is believed to exist in Nature, so also are we bound to admit virtual extension ; 

& if this is admitted, then it will be possible for the primary elements of matter to be 

simple, & yet not absolutely non-extended. 

84. But I have come to the conclusion that this idea is quite overthrown by that same virtual extension 
principle of induction, by which we have hitherto deduced so many results which we have is r .excluded^ by the 
employed. For we see, in all those bodies that we can bring under observation, that auction 6 correctly 
whatever occupies a distinct position is itself also a distinct thing ; so that those that occupy a PP lied - 
different parts of space can be separated by using a sufficiently large force ; nor can we 

detect a case in which these larger bodies have any part that occupies different parts of 
space at one & the same time, & yet is the same part. Further, this property by its very 
nature is of the sort for which it is equally probable that it happens in magnitudes that we can 
detect by the senses & in magnitudes which are below the limits of our senses. In fact, 
the matter depends only upon the size of the space, throughout which the virtual extension 
is supposed to exist ; & this size, if it were sufficiently ample, would become sensible 
to us. Since then we never find this virtual extension in magnitudes that fall within the 
range of our senses, nay rather, in innumerable cases we perceive the contrary ; the matter 
certainly ought to be transferred by the principle of induction, as explained above, to 
any of the smallest particles of matter as well ; so that not even they are admitted to have 
such virtual extension. 

85. The illustrations that are added, derived from a consideration of the rational Reply to the 
soul & the omnipresence of GOD, prove nothing positively ; for they are derived from s^uf&'cot) 6 ' 
another class of entities, except that, I do not think that it can even be proved that the 

rational soul does not exist in merely a single, simple, & non-extended point of the body ; 
so that it maintains the same position, & thence it puts forth some sort of force into the 
remaining points of the body duly disposed about it ; & the intercommunication between 
the soul & the body consists of these forces, some of which are involuntary whilst others 
are voluntary. Further, we are absolutely ignorant of the nature of the presence of GOD ; 
& in no wise do we say that He is really extended throughout divisible space ; nor from 
those modes, surpassing all human intelligence, by which HE exists, thinks, wills & acts, 
can any analogy or deduction be made which will apply to human or material modes of 
existence & action. 

86. Again, as regards the analogy with rest, we have arguments that are sufficiently Again with regard 

IT T i j i i i_ t ^v vr .. ' to the analogy with 

strong to lead us to believe, as I remarked above, that there is no such thing m Nature rest . 
as absolute rest. Indeed, I proved that such a thing could not be, by a direct argument 


existere, argumento quodam positive ex numero combinationum possibilium infinite 
contra alium finitum, demonstravi in Stayanis Supplementis, ubi de spatio, & tempore 
quae juxta num. 66 occurrent infra Supplementorum i, & 2 ; numquam vero earn 
existere in Natura, patet sane in ipsa Newtoniana sententia de gravitate generali, in qua in 
planetario systemate ex mutuis actionibus quiescit tantummodo centrum commune gravi- 
tatis, punctum utique imaginarium, circa quod omnia planetarum, cometarumque corpora 
moventur, ut & ipse Sol ; ac idem accidit fixis omnibus circa suorum systematum gravitatis 
centra ; quin immo ex actione unius systematis in aliud utcunque distans, in ipsa gravitatis 
centra motus aliquis inducetur ; & generalius, dum movetur quaecunque materiae particula, 
uti luminis particula qusecunque ; reliquae omnes utcunque remotae, quas inde positionem 
ab ilia mutant, mutant & gravitatem, ac proinde moventur motu aliquo exiguo, sed sane 
motu. In ipsa Telluris quiescentis sententia, quiescit quidem Tellus ad sensum, nee tota 
ab uno in alium transfertur locum ; at ad quamcunque crispationem maris, rivuli decursum, 
muscae volatum, asquilibrio dempto, trepidatio oritur, perquam exigua ilia quidem, sed 
ejusmodi, ut veram quietem omnino impediat. Quamobrem analogia inde petita evertit 
potius virtualem ejusmodi simplicium elementorum extensionem positam in conjunctione 
ejusdem momenti temporis cum serie continua punctorum loci, quam comprobet. 

in quo deficiat ana- 87. Sed nee ea ipsa analogia, si adesset, rem satis evinceret ; cum analogiam inter tempus, 

logia loci, & tem- l ocum videamus in aliis etiam violari : nam in iis itidem paragraphis Supplementorum 
demonstravi, nullum materiae punctum unquam redire ad punctum spatii quodcunque, 
in quo semel fuerit aliud materiae punctum, ut idcirco duo puncta materiae nunquam 
conjungant idem [40] punctum spatii ne cum binis quidem punctis temporis, dum quam- 
plurima binaria punctorum materiae conjungunt idem punctum temporis cum duobus 
punctis loci ; nam utique coexistunt : ac praeterea tempus quidem unicam dimensionem 
habet diuturnitatis, spatium vero habet triplicem, in longum, latum, atque profundum. 

inextensio utilis 88. Quamobrem illud jam tuto inferri potest, haec primigenia materiae elementa, non 

ad exciudendum so i um esse s i m pli c ia, ac indivisibilia, sed etiam inextensa. Et quidem haec ipsa simplicitas, 

transitum momen- , t ; . i i_ j ji_ 

taneum a densitate & inextensio elementorum praestabit commoda sane plunma, quibus eadem adnuc magis 
nuiia ad summam. f u i c itur, ac comprobatur. Si enim prima elementa materiae sint quaedam partes solidse, 
ex partibus compositae, vel etiam tantummodo extensae virtualiter, dum a vacuo spatio 
motu continue pergitur per unam ejusmodi particulam, fit saltus quidam momentaneus 
a densitate nulla, quae habetur in vacuo, ad densitatem summam, quae habetur, ubi ea 
particula spatium occupat totum. Is vero saltus non habetur, si elementa simplicia sint, 
& inextensa, ac a se invicem distantia. Turn enim omne continuum est vacuum tantum- 
modo, & in motu continue per punctum simplex fit transitus a vacuo continue ad vacuum 
continuum. Punctum illud materiae occupat unicum spatii punctum, quod punctum 
spatii est indivisibilis limes inter spatium praecedens, & consequens. Per ipsum non 
immoratur mobile continue motu delatum, nee ad ipsum transit ab ullo ipsi immediate 
proximo spatii puncto, cum punctum puncto proximum, uti supra diximus, nullum sit ; 
sed a vacuo continue ad vacuum continuum transitur per ipsum spatii punctum a materiae 
puncto occupatum. 

itidem ad hoc, ut go,. Accedit, quod in sententia solidorum, extensorumque elementorum habetur illud, 

possit, ut p"test densitatem corporis minui posse in infinitum, augeri autem non posse, nisi ad certum limitem 
minui in infinitum. i n q uo increment! lex necessario abrumpi debeat. Primum constat ex eo, quod eadem 
particula continua dividi possit in particulas minores quotcunque, quae idcirco per spatium 
utcunque magnum diffundi potest ita, ut nulla earum sit, quae aliquam aliam non habeat 
utcunque libuerit parum a se distantem. Atque eo pacto aucta mole, per quam 
eadem ilia massa diffusa sit, eaque aucta in ratione quacunque minuetur utique 
densitas in ratione itidem utcunque magna. Patet & alterum : ubi enim omnes 
particulae ad contactum devenerint ; densitas ultra augeri non poterit. Quoniam 
autem determinata quaedam erit utique ratio spatii vacui ad plenum, nonnisi in ea ratione 
augeri poterit densitas, cujus augmentum, ubi ad contactum deventum fuerit, adrumpetur. 
At si elementa sint puncta penitus indivisibilia, & inextensa ; uti augeri eorum distantia 
poterit in infinitum, ita utique poterit etiam minui pariter in ratione quacunque ; cum 


founded upon the infiniteness of a number of possible combinations as against the finiteness 
of another number, in the Supplements to Stay's Philosophy, in connection with space 
& time ; these will be found later immediately after Art. 14 of the Supplements, I 
and II. That it never does exist in Nature is really clear in the Newtonian theory of 
universal gravitation ; according to this theory, in the planetary system the common centre 
of gravity alone is at rest under the action of the mutual forces ; & this is an altogether 
imaginary point, about which all the bodies of the planets & comets move, as also does 
the sun itself. Moreover the same thing happens in the case of all the fixed stars with regard 
to the centres of gravity of their systems ; & from the action of one system on another 
at any distance whatever from it, some motion will be imparted to these very centres of 
gravity. More generally, so long as any particle of matter, so long as any particle of light, 
is in motion, all other particles, no matter how distant, which on account of this motion 
have their distance from the first particle altered, must also have their gravitation altered, 
& consequently must move with some very slight motion, but yet a true motion. In 
the idea of a quiescent Earth, the Earth is at rest approximately, nor is it as a whole translated 
from place to place ; but, due to any tremulous motion of the sea, the downward course 
of rivers, even to the fly's flight, equilibrium is destroyed & some agitation is produced, 
although in truth it is very slight ; yet it is quite enough to prevent true rest altogether. 
Hence an analogy deduced from rest contradicts rather than corroborates virtual extension 
of the simple elements of Nature, on the hypothesis of a conjunction of the same instant 
of time with a continuous series of points of space. 

87. But even if the foregoing analogy held good, it would not prove the matter Where the analogy 
satisfactorily ; since we see that in other ways the analogy between space & time is impaired. 2^ pace and tlme 
For I proved, also in those paragraphs of the Supplements that I have mentioned, that 

no point of matter ever returned to any point of space, in which there had once been any 
other point of matter ; so that two points of matter never connected the same point of 
space with two instants of time, let alone with more ; whereas a huge number of pairs of 
points connect the same instant of time with two points of space, since they certainly coexist. 
Besides, time has but one dimension, duration ; whilst space has three, length, breadth 
& depth. 

88. Therefore it can now be safely accepted that these primary elements of matter Non-extension use- 
are not only simple & indivisible, but also that they are non-extended. Indeed this a u n \nstanTaneous 
very simplicity & non-extension of the elements will prove useful in a really large number passage from no 
of cases for still further strengthening & corroborating the results already obtained. J^-one. a Very 
For if the primary elements were certain solid parts, themselves composed of parts or even 

virtually extended only, then, whilst we pass by a continuous motion from empty space 
through one particle of this kind, there would be a sudden change from a density that is 
nothing when the space is empty, to a density that is very great when the particle occupies 
the whole of the space. But there is not this sudden change if we assume that the elements 
are simple, non-extended & non-adjacent. For then the whole of space is merely a 
continuous vacuum, &, in the continuous motion by a simple point, the passage is made 
from continuous vacuum to continuous vacuum. The one point of matter occupies but 
one point of space ; & this point of space is the indivisible boundary between the space 
that precedes & the space that follows. There is nothing to prevent the moving point 
from being carried through it by a continuous motion, nor from passing to it from any 
point of space that is in immediate proximity to it : for, as I remarked above, there 
is no point that is the next point to a given point. But from continuous vacuum 
to continuous vacuum the passage is made through that point of space which is occupied 
by the point of matter. 

89. There is also the point, that arises in the theory of solid extended elements, namely Also for the idea 
that the density of a body can be diminished indefinitely, but cannot be increased except j^^a'^ ^can 
up to a certain fixed limit, at which the law of increase must be discontinuous. The first be decreased, 
comes from the fact that this same continuous particle can be divided into any number mdefinltel y- 

of smaller particles ; these can be diffused through space of any size in such a way that 
there is not one of them that does not have some other one at some little (as little as you 
will) distance from itself. In this way the volume through which the same mass is diffused 
is increased ; & when that is increased in any ratio whatever, then indeed the density 
will be diminished in the same ratio, no matter how great the ratio may be. The second 
thing is also evident ; for when the particles have come into contact, the density cannot 
be increased any further. Moreover, since there will undoubtedly be a certain determinate 
ratio for the amount of space that is empty compared with the amount of space that is 
full, the density can only be increased in that ratio ; & the regular increase of density 
will be arrested when contact is attained. But if the elements are points that are perfectly 
indivisible & non-extended, then, just as their distances can be increased indefinitely, 


in [41] ratione quacunque lineola quaecunque secari sane possit : adeoque uli nullus est 
limes raritatis auctae, ita etiam nullus erit auctae densitatis. 

Et ad excludendum 9- Sed & illud commodum accidet, quod ita omne continuum coexistens eliminabitur 

continuum extcn- e Natura, in quo explicando usque adeo dcsudarunt, & fere incassum, Philosophi, ncc idcirco 

sum, & in infinitum j ** r j i i 

in existentibus. divisio ulla realis entis in innmtum produci potent, nee naerebitur, ubi quaeratur, an numerus 
partium actu distinctarum, & separabilium, sit finitus, an infinitus ; nee alia ejusmodi 
sane innumera, quae in continui compositione usque adeo negotium facessunt Philosophis, 
jam habebuntur. Si enim prima materiae elementa sint puncta penitus inextensa, & 
indivisibilia, a se invicem aliquo intervallo disjuncta ; jam erit finitus punctorum numerus 
in quavis massa : nam distantiae omnes finitae erunt ; infinitesimas enim quantitates in se 
determinatas nullas esse, satis ego quidem, ut arbitror, luculcnter demonstravi & in disser- 
tatione De Natura, t$ Usu infinitorum, ac infinite parvorum, & in dissertatione DC Lege 
Continuitatis, & alibi. Intervallum quodcunque finitum erit, & divisibile utique in 
infinitum per interpositionem aliorum, atque aliorum punctorum, quae tamen singula, 
ubi fuerint posita, finita itidem erunt, & aliis pluribus, finitis tamen itidem, ubi extiterint, 
locum reliquent, ut infinitum sit tantummodo in possibilibus, non autem in existentibus, 
in quibus possibilibus ipsis omnem possibilium seriem idcirco ego appellare soleo constantem 
terminis finitis in infinitum, quod quaecunque, quae existant, finita esse debeant, sed nullus 
sit existentium finitus numerus ita ingens, ut alii, & alii majores, sed itidem finiti, haberi 
non possint, atque id sine ullo limite, qui nequeat praeteriri. Hoc autcm pacto, sublato 
ex existentibus omni actuali infinite, innumerae sane difficultates auferentur. 

inextensionem 91. Cum igitur & positive argumento. a lege virium positive demonstrata desumpto, 

qua'rend^m^e simplicitas, & inextensio primorum materiae elementorum deducatur, tam multis aliis 

homogeneitate. vel indiciis fulciatur, vel emolumentis inde derivatis confirmetur ; ipsa itidem admitti 

jam debet, ac supererit quaerendum illud tantummodo, utrum haec elementa homogenca 

censeri debeant, & inter se prorsus similia, ut ea initio assumpsimus, an vero heterogenea, 

ac dissimilia. 

Homogeneitatem 92. Pro homogeneitate primorum materiae elementorum illud est quoddani veluti 

genefta 1 te a primi! n( & P rm cipium, quod in simplicitate, & inextensione conveniant, ac etiam vires quasdam habeant 
uitimi asymptotici utique omnia. Deinde curvam ipsam virium eandem esse omnino in omnibus illud indicat, 
omnibus' P "' S ve ^ e tiani evincit, quod primum crus repulsivum impenetrabilitatem secum trahens, & 
postremum attractivum gravitatem definiens, omnino communia in omnibus sint : nam 
corpora omnia aeque impenetrabilia sunt, & vero etiam aeque gravia pro quantitate materiae 
suae, uti satis [42] evincit aequalis velocitas auri, & plumse cadentis in Boyliano recipiente 
Si reliquus curvae arcus intermedius esset difformis in diversis materiae punctis ; infinities 
probabilius esset, difformitatem extendi etiam ad crus primum, & ultimum, cum infinities 
plures sint curvae, quae, cum in reliquis differant partibus, differant plurimum etiam in 
hisce extremis, quam quae in hisce extremis tantum modo tam arete consentiant. Et hoc 
quidem argumento illud etiam colligitur, curvam virium in quavis directione ab eodem 
primo materiae elemento, nimirum ab eodem materiae puncto eandem esse, cum & primum 
impenetrabilitatis, & postremum gravitatis crus pro omnibus directionibus sit ad sensum 
idem. Cum primum in dissertatione De Firibus Vivis hanc Theoriam protuli, suspicabar 
diversitatem legis ' virium respondentis diversis directionibus ; sed hoc argumento adi 
majorem simplicitatem, & uniformitatem deinde adductus sum. Diversitas autem legum 
virium pro diversis particulis, & pro diversis respectu ejusdem particulae directionibus, 
habetur utique ex diverso numero, & positione punctorum earn componentium, qua de 
re inferius aliquid. 

i contra deduci 93- Nee vero huic homogeneitati opponitur inductionis principium, quo ipsam 
ex principio indis- Leibnitiani oppugnare solent, nee principium rationis sufficients, atque indiscernibilium, 

cermbUium, & rati- . . T TV- /" j- -j 

onis sufficients. quod supenus innui numero 3. Innmtam Divini v_onditons mentem, ego quidem omnino. 
arbitror, quod & tam multi Philosophi censuerunt, ejusmodi perspicacitatem habere, atque 
intuitionem quandam, ut ipsam etiam, quam individuationem appellant, omnino similium 
individuorum cognoscat, atque ilia inter se omnino discernat. Rationis autem sufficientis 


so also can they just as well be diminished in any ratio whatever. For it is certainly possible 
that a short line can be divided into parts in any ratio whatever ; & thus, just as there 
is no limit to increase of rarity, so also there is none to increase of density. 

qo. The theory of non-extension is also convenient for eliminating from Nature all ^ lso -/ or excludm s 

7 / 1 1 1 1 ! 1 1 Ml 11 1 6 *" ea a C011 " 

idea of a coexistent continuum to explain which philosophers have up till now laboured tinuum in existing 

so very hard & generally in vain. Assuming non-extension, no division of a real entity thm R s - that 

can be carried on indefinitely ; we shall not be brought to a standstill when we seek to 
find out whether the number of parts that are actually distinct & separable is finite or 
infinite ; nor with it will there come in any of those other truly innumerable difficulties 
that, with the idea of continuous composition, have given so much trouble to philosophers. 
For if the primary elements of matter are perfectly non-extended & indivisible points 
separated from one another by some definite interval, then the number of points in any 
given mass must be finite ; because all the distances are finite. I proved clearly enough, 
I think, in the dissertation De Natura, & Usu infinitorum ac infinite parvorum, & in the 
dissertation De Lege Continuitatis, & in other places, that there are no infinitesimal 
quantities determinate in themselves. Any interval whatever will be finite, & at least 
divisible indefinitely by the interpolation of other points, & still others ; each such set 
however, when they have been interpolated, will be also finite in number, & leave room 
for still more ; & these too, when they existed, will also be finite in number. So that 
there is only an infinity of possible points, but not of existing points ; & with regard 
to these possible points, I usually term the whole series of possibles a series that ends at 
finite limits at infinity. This for the reason that any of them that exist must be finite 
in number ; but there is no finite number of things that exist so great that other numbers, 
greater & greater still, but yet all finite, cannot be obtained ; & that too without any 
limit, which cannot be surpassed. Further, in this way, by doing away with all idea of 
an actual infinity in existing things, truly countless difficulties are got rid of. 

91. Since therefore, by a direct argument derived from a law of forces that has been Non-extension 
directly proved, we have both deduced the simplicity & non-extension of the primary w" 5 have a now e to 
elements of matter, & also we have strengthened the theory by evidence pointing towards investigate homo- 
it, or corroborated it by referring to the advantages to be derived from it ; this theory gen 

ought now to be accepted as true. There only remains the investigation as to whether 
these elements ought to be considered to be homogeneous & perfectly similar to one 
another, as we assumed at the start, or whether they are really heterogeneous & dissimilar. 

92. In favour of the homogeneity of the primary elements of matter we have so to Homogeneity for 
speak some foundation derived from the fact that all of them agree in simplicity & non- V oca!ted St f rom a a 
extension, & also that they are all endowed with forces of some sort. Now, that this consideration of 
curve of forces is exactly the same for all of them is indicated or even proved by the fact O f 6 the fir?t 86 last 
that the first repulsive branch necessitating impenetrability, & the last attractive branch a s y"m p t o t i c 
determining gravitation, are exactly the same in all respects. For all bodies are equally c ^l S forces* 
impenetrable ; & also all are equally heavy in proportion to the amount of matter 

contained in them, as is sufficiently proved by the equal velocity of the piece of gold & 
the feather when falling in Boyle's experiment. If the remaining intermediate arc of the 
curve were non-uniform for different points of matter, it would be infinitely more probable 
that the non-uniformity would extend also to the first & last branches also ; for there 
are infinitely more curves which, when they differ in the remaining parts, also differ to 
the greatest extent in the extremes, than there are curves, which agree so closely only in 
these extremes. Also from this argument we can deduce that the curve of forces is indeed 
exactly the same from the same point of matter, in any direction whatever from the same 
primary element of matter ; for both the first branch of impenetrability & the last branch 
of gravitation are the same, so far as we can perceive, for all directions. When I first 
published this Theory in my dissertation De Firibus Fivis, I was inclined to believe that 
there was a diversity in the law of forces corresponding to diversity of direction ; but I 
was led by the argument given above to the greater simplicity & the greater uniformity 
derived therefrom. Further, diversity of the laws of forces for diverse particles, & for 
different directions with the same particle, is certainly to be obtained from the diverse 
number & position of the points composing it ; about which I shall have something 
to say later. 

93. Nor indeed is there anything opposed to this idea of homogeneity to be derived Notl ?i n s t? b 

r i i r i J i o rr t o /_ . brought against 

from the principle of induction, by means of which the followers of Leibniz usually raise this from the doc- 
an objection to it ; nor from the principle of sufficient reason, & of indiscernibles, that fc es f . ind j. scern : 

J . , . . . -r r i i -TO i_ ibles & sufficient 

1 mentioned above in Art. 3. I am indeed quite convinced, & a great many other reason. 1 
philosophers too have thought, that the Infinite Will of the Divine Founder has a 
perspicacity & an intuition of such a nature that it takes cognizance of that which is 
called individuation amongst individuals that are perfectly similar, & absolutely 



principium falsum omnino esse censeo, ac ejusmodi, ut omnem verse libertatis ideam omnino 
tollat ; nisi pro ratione, ubi agitur de voluntatis determinatione, ipsum liberum arbitrium, 
ipsa libera determinatio assumatur, quod nisi fiat in voluntate divina, quaccunque existunt, 
necessario existunt, & qusecunque non existunt, ne possibilia quidem erunt, vera aliqua 
possibilitate, uti facile admodum demonstratur ; quod tamen si semel admittatur, mirum 
sane, quam prona demum ad fatalem necessitatem patebit via. Quamobrem potest divina 
voluntas determinari ex toto solo arbitrio suo ad creandum hoc individuum potius, quam 
illud ex omnibus omnino similibus, & ad ponendum quodlibet ex iis potius eo loco, quo 
ponit, quam loco alterius. Sed de rationis sufficientis principio haec ipsa fusius pertractavi 
turn in aliis locis pluribus, turn in Stayanis Supplementis, ubi etiam illud ostendi, id prin- 
cipium nullum habere usum posse in iis ipsis casibus, in quibus adhibetur, & praedicari solet 
tantopere, atque id idcirco, quod nobis non innotescant rationes omnes, quas tamen 
oporteret utique omnes nosse ad hoc, ut eo principio uti possemus, amrmando, nullam 
esse rationem sufncientem pro hoc potius, quam pro illo [43] alio : sane in exemplo illo 
ipso, quod adhiberi solet, Archimedis hoc principio aequilibrium determinantis, ibidem 
ostendi, ex ignoratione causarum, sive rationum, quse postea detectae sunt, ipsum in suae 
investigationis progressu errasse plurimum, deducendo per abusum ejus principii sphsericam 
figuram marium, ac Telluris. 


Posse etiam puncta 94. Accedit & illud, quod ilia puncta materiae, licet essent prorsus similia in simplicitate, 

dlfierrTin aiiis 11S> & extensione, ac mensura virium, pendentium a distantia, possent alias habere proprietates 

metaphysicas diversas inter se, nobis ignotas, quae ipsa etiam apud ipsos Leibnitianos 


Non vaierehicprin- 95. Quod autem attinet ad inductionem, quam Leibnitiani desumunt a dissimilitudine, 

a^ma^sis^eas^de! quam observamus in rebus omnibus, cum nimirum nusquam ex. gr. in amplissima silva reperire 
ferre ex diversis sit duo folia prorsus similia ; ea sane me nihil movet ; cum nimirum illud discrimen sit 
p rO pri e tas relativa ad rationem aggregati, & nostros sensus, quos singula materiae elementa 
non afficiunt vi sufficiente ad excitandam in animo ideam, nisi multa sint simul, & in molem 
majorem excrescant. Porro scimus utique combinationes ejusdem numeri terminorum 
in immensum excrescere, si ille ipse numerus sit aliquanto major. Solis 24 litterulis 
Alphabet! diversimodo combinatis formantur voces omnes, quibus hue usque usa sunt 
omnia idiomata, quae extiterunt, & quibus omnia ilia, quae possunt existere, uti possunt. 
Quid si numerus earum existeret tanto major, quanto major est numerus puuctorum 
materiae in quavis massa sensibili ? Quod ibi diversus est litterarum diversarum ordo, id 
in punctis etiam prorsus homogeneis sunt positiones, & distantia, quibus variatis, variatur 
utique forma, & vis, qua sensus afficitur in aggregatis. Quanto major est numerus 
combinationum diversarum possibilium in massis sensibilibus, quam earum massarum, quas 
possumus observare, & inter se conferre (qui quidem ob distantias, & directiones in infinitum 
variabiles praescindendo ab aequilibrio virium, est infinitus, cum ipso aequilibrio est immen- 
sus) ; tanto major est improbabilitas duarum massarum omnino similium, quam omnium 
aliquantisper saltern inter se dissimilium. 

Physica ratio dis- 96. Et quidem accedit illud etiam, quod alicujus dissimilitudinis in aggregatis physicam 

massU 1 ut 1 in 1 fo r iiu US I 1100 ! 116 rationem cernimus in iis etiam casibus, in quibus maxime inter se similia esse 
deberent. Cum enim mutuae vires ad distantias quascunque pertineant ; status uniuscu- 
jusque puncti pendebit saltern aliquantisper a statu omnium aliorum punctorum, quae 
sunt in Mundo. Porro utcunque puncta quaedam sint parum a se invicem remota, uti 
sunt duo folia in eadem silva, & multo magis in eodem ramo ; adhuc tamen non eandem 
prorsus relationem distantiae, & virium habent ad reliqua omnia materiae puncta, quae 
[44] sunt in Mundo, cum non eundem prorsus locum obtineant ; & inde jam in aggregate 
discrimen aliquod oriri debet, quod perfectam similitudinem omnino impediat. Sed illud 
earn inducit magis, quod quae maxime conferunt ad ejusmodi dispositionem, necessario 
respectu diversarum frondium diversa non nihil esse debeant. Omissa ipsa earum forma 
in semine, solares radii, humoris ad nutritionem necessarii quantitas, distantia, a qua debet 
is progredi, ut ad locum suum deveniat, aura ipsa, & agitatio inde orta, non sunt omnino 
similia, sed diversitatem aliquam habent, ex qua diversitas in massas inde efformatas 


distinguishes them one from the other. Moreover, I consider that the principle of sufficient 
reason is altogether false, & one that is calculated to take away all idea of true freewill. 
Unless free choice or free determination is assumed as the basis of argument, in discussing 
the determination of will, unless this is the case with the Divine Will, then, whatever 
things exist, exist because they must do so, & whatever things do not exist will not even 
be possible, i.e., with any real possibility, as is very easily proved. Nevertheless, once this 
idea is accepted, it is truly wonderful how it tends to point the way finally to fatalistic 
necessity. Hence the Divine Will is able, of its own pleasure alone, to be determined 
to the creation of one individual rather than another out of a whole set of exactly similar 
things, & to the setting of any one of these in the place in which it puts it rather than in 
the place of another. But I have discussed these very matters more at length, besides several 
other places, in the Supplements to Stay's Philosophy ; where I have shown that the 
principle cannot be employed in those instances in which it is used & generally so strongly 
asserted. The reason being that all possible reasons are not known to us ; & yet they 
should certainly be known, to enable us to employ the principle by stating that there is 
no sufficient reason in favour of this rather than that other. In truth, in that very example 
of the principle generally given, namely, that of Archimedes' determination of equilibrium 
by means of it, I showed also that Archimedes himself had made a very big mistake in following 
out his investigation because of his lack of knowledge of causes or reasons that were discovered 
in later days, when he deduced a spherical figure for the seas & the Earth by an abuse 
of this principle. 

94. There is also this, that these points of matter, although they might be perfectly it is possible for 
similar as regards simplicity & extension, & in having the measure of their forces depen- ^"^ese^ro erties 
dent on their distances, might still have other metaphysical properties different from one but to disagree in 
another, & unknown to us ; & these distinctions also are made by the followers of others - 

95. As regards the induction which the followers of Leibniz make from the lack of The principle does 
similitude that we see in all things, (for instance such as that there never can be found in n t . hold g d here 

T_ i j i i vi \ i i . , * induction from 

the largest wood two leaves exactly alike), their argument does not impress me in the masses; they differ 

slightest degree. For that distinction is a property that is concerned with reasoning for .n account of 

an aggregate, & also with our senses ; & these senses single elements of matter cannot tionsof their parta. 

influence with sufficient force to excite an idea in the mind, except when there are many 

of them together at a time, & they develop into a mass of considerable size. Further 

it is well known that combinations of the same number of terms increase enormously, if 

that number itself increase a little. From the 24 letters of the alphabet alone, grouped 

together in different ways, are formed all the words that have hitherto been used in all 

expressions that have existed, or can possibly come into existence. What then if their 

number were increased to equal the number of points of matter in any sensible mass ? 

Corresponding to the different order of the several letters in the one, we have in perfectly 

homogeneous points also different positions & distances ; & if these are altered at least 

the form & the force, which affect our senses in the groups, are altered as well. How 

much greater is the number of different combinations that are possible in sensible masses 

than the number of those masses that we can observe & compare with one another (& 

this number, on account of the infinitely variable distances & directions of the forces, 

when equilibrium is precluded, is infinite, since including equilibrium it is very great) ; 

just so much greater is the improbability of two masses being exactly similar than of 

their being all at least slightly different from one another. 

96. There is also this point in addition ; we discern a physical reason as well for some Physical reason for 
dissimilarity in groups for those cases too, in which they ought to be especially similar to the difference in 

.1 -n i f 11 -11 T r i ' several masses, as 

one another, ror since mutual forces pertain to all possible distances, the state of any in leaves. 
one point will depend upon, at least in some slight degree, the state of all other points 
that are in the universe. Further, however short the distance between certain points may 
be, as of two leaves in the same wood, much more so on the same branch, still for all 
that they do not have quite the same relation as regards distance & forces as all the rest 
of the points of matter that are in the universe, because they do not occupy quite the 
same place. Hence in a group some distinction is bound to arise which will entirely prevent 
perfect similarity. Moreover this tendency is all the stronger, because those things which 
especially conduce to this sort of disposition must necessarily be somewhat different with 
regard to different leaves. For the form itself being absent in the seed, the rays of the 
sun, the quantity of moisture necessary for nutrition, the distance from which it has to 
proceed to arrive at the place it occupies, the air itself & the continual motion derived 
from this, these are not exactly similar, but have some diversity ; & from this diversity 
there proceeds a diversity in the masses thus formed. 


simiiitudine quaii- 97. Patet igitur, varietatem illam a numero pendere combinationum possibilium in 

^ numero punctorum necessario ad sensationem, & circumstantiarum, quae ad formationem 

geneitatem, quam massze sunt neccssariae, adeoque ejusmodi inductionem extend! ad elementa non posse. 

* ' 

Q u i n immo ilia tanta similitude, quae cum exigua dissimilitudine commixta invenitur in 
tarn multis corporibus, indicat potius similitudinem ingentem in elementis. Nam ob 
tantum possibilium combinationum numerum, massae elementorum etiam penitus homo- 
geneorum debent a se invicem differre plurimum, adeoque si elementa heterogenea sint, 
in immensum majorem debent habere dissimilitudinem, quam ipsa prima elementa, ex 
quibus idcirco nullae massas, ne tantillum quidem, similes provenire deberent. Cum 
elementa multo minus dissimilia esse debeant, quam aggregata elementorum, multo 
magis ad elementorum homogeneitatem valere debet ilia quaecunque similitudo, quam 
in corporibus observamus, potissimum in tarn multis, quae ad eandem pertinent speciem, 
quam ad homogeneitatem eorundem tarn exiguum illud discrimen, quod in aliis tarn 
multis observatur. Rem autem penitus conficit ilia tanta similitudo, qua superius usi 
sumus, in primo crure exhibente impenetrabilitatem, & in postremo exhibente gravitatem 
generalem, quae crura cum ob hasce proprietates corporibus omnibus adeo generales, adeo 
inter se in omnibus similia sint, etiam reliqui arcus curvae exprimentis vires omnimodam 
similitudinem indicant pro corporibus itidem omnibus. 

Homogeneitatem 98. Superest, quod ad hanc rem pertinet, illud unum iterum hie monendum, quod 

insinuarr' ^xem* ip sum etiam initio hujus Operis innui, ipsam Naturam, & ipsum analyseos ordinem nos 

plum a libris, lit- ducere ad simplicitatem & homogeneitatem elementorum, cum nimirum, quo analysis 

ns> pul promovetur magis, eo ad pauciora, & inter se minus discrepantia principia deveniatur, uti 

patet in resolutionibus Chemicis. Quam quidem rem ipsum litterarum, & vocum exemplum 

multo melius animo sistet. Fieri utique possent nigricantes litteras, non ductu atramenti 

continue, sed punctulis rotundis nigricantibus, & ita parum a se invicem remotis, ut inter- 

valla non nisi ope microscopii discerni possent, & quidem ipsae litterarum formae pro typis 

fieri pos-[45]-sent ex ejusmodi rotundis sibi proximis cuspidibus constantes. Concipiatur 

ingens quaedam bibliotheca, cujus omnes libri constent litteris impressis, ac sit incredibilis 

in ea multitude librorum conscriptorum linguis variis, in quibus omnibus forma charac- 

terum sit eadem. Si quis scripturae ejusmodi, & linguarum ignarus circa ejusmodi libros, 

quos omnes a se invicem discrepantes intueretur, observationem institueret cum diligenti 

contemplatione ; primo quidem inveniret vocum farraginem quandam, quae voces in 

quibusdam libris occurrerent saepe, cum eaedem in aliis nusquam apparent, & inde lexica 

posset quaedam componere totidem numero, quot idiomata sunt, in quibus singulis omnes 

ejusdem idiomatis voces reperirentur, quae quidem numero admodum pauca essent, discri- 

mine illo ingenti tot, tarn variorum librorum redacto ad illud usque adeo minus discrimen, 

quod contineretur lexicis illis, & haberetur in vocibus ipsa lexica constituentibus. At 

inquisitione promota, facile adverteret, omnes illas tarn varias voces constare ex 24 

tantummodo diversis litteris, discrimen aliquod inter se habentibus in ductu linearum, 

quibus formantur, quarum combinatio diversa pareret omnes illas voces tarn varias, ut 

earum combinatio libros efformaret usque adeo magis a se invicem discrepantes. Et ille 

quidem si aliud quodcunque sine microscopic examen institueret, nullum aliud inveniret 

magis adhuc simile elementorum genus, ex quibus diversa ratione combinatis orirentur 

ipsae litterse ; at microscopic arrepto, intueretur utique illam ipsam litterarum composi- 

tionem e punctis illis rotundis prorsus homogeneis, quorum sola diversa positio, ac 

distributio litteras exhiberet. 

Appiicatio exempli 99. Haec mihi quaedam imago videtur esse eorum, quae cernimus in Natura. Tarn 

a<^ Naturae analy- mu \ t { } tam var ;j fift ijb r j corpora sunt, & quae ad diversa pertinent regna, sunt tanquam 
diversis conscripta linguis. Horum omnium Chemica analysis principia quaedam invenit 
minus inter se difrormia, quam sint libri, nimirum voces. Hae tamen ipsae inter se habent 
discrimen aliquod, ut tam multas oleorum, terrarum, salium species eruit Chemica analysis 
e diversis corporibus. Ulterior analysis harum, veluti vocum, litteras minus adhuc inter 
se difformes inveniret, & ultima juxta Theoriam meam deveniret ad homogenea punctula, 
quae ut illi circuli nigri litteras, ita ipsa diversas diversorum corporum particulas per solam 
dispositionem diversam efformarent : usque adeo analogia ex ipsa Naturae consideratione 


97. It is clear then that this variety depends on the number of possible combinations Homogeneity is to 
to be found for the number of points that are necessary to make the mass sensible, & ^ m d< ^ ort * ot 
of the circumstances that arenecessary for the formation of the mass ; & so it is not similitude in some 
possible that the induction should be extended to the elements. Nay rather, the great heterogeneity from" 
similarity that is found accompanied by some very slight dissimilarity in so many bodies dissimilarity. 
points more strongly to the greatest possible similarity of the elements. For on account 

of the great number of the possible combinations, even masses of elements that are perfectly 
homogeneous must be greatly different from one another ; & thus if the elements are 
heterogeneous, the masses must have an immensely greater dissimilarity than the primary 
elements themselves ; & therefore no masses formed from these ought to come out similar, 
not even in the very slightest degree. Since the elements are bound to be much less 
dissimilar than aggregates formed from these elements, homogeneity of the elements must 
be indicated by that certain similarity that we observe in bodies, especially in so many 
of those that belong to the same species, far more strongly than heterogeneity of the elements 
is indicated by the slight differences that are observed in so many others. The whole 
discussion is made perfectly complete by that great similarity, which we made use of above, 
that exists in the first branch representing impenetrability, & in the last branch representing 
universal gravitation ; for since these branches, on account of properties that are so general 
to all bodies, are so similar to one another in all cases, they indicate complete similarity 
of the remaining arc of the curve expressing the forces for all bodies as well. 

98. Naught that concerns this subject remains but for me to once more mention in Homogeneity is 
this connection that one thing, which I have already remarked at the beginning of this anftysis of Nature" 
work, namely, that Nature itself & the method of analysis lead us towards simplicity & example taken 
homogeneity of the elements ; since in truth the farther the analysis is pushed, the fewer anc j dots ' 

the fundamental substances we arrive at & the less they differ from one another ; as is 
to be seen in chemical experiments. This will be presented to the mind far more clearly 
by an illustration derived from letters & words. Suppose we have made black letters, 
not by drawing a continuous line with ink, but by means of little black dots which are at 
such small distances from one another that the intervals cannot be perceived except with 
the aid of a microscope & indeed such forms of letters may be made as types from round 
points of this sort set close to one another. Now imagine that we have a huge library, 
all the books in it consisting of printed letters, & let there be an incredible multitude 
of books printed in various languages, in all which the form of the characters is the same. 
If anyone, who was ignorant of such compositions or languages, started on a careful study 
of books of this kind, all of which he would perceive differed from one another ; then first 
of all he would find a medley of words, some of which occurred frequently in certain books 
whilst they never appeared at all in others. Hence he could compose lexicons, as many 
in number as there are languages ; in each of these all words of the same language would 
be found, & these would indeed be very few in number ; for the immense multiplicity 
of words in this numerous collection of books of so many kinds is now reduced to what 
is still a multiplicity, but smaller, than is contained in the lexicons & the words forming 
these lexicons. Now if he continued his investigation, he would easily perceive that the 
whole of these words of so many different kinds were formed from 24 letters only ; that 
these differed in some sort from one another in the manner in which the lines forming 
them were drawn ; that the different combinations of these would produce the whole of 
that great variety of words, & that combinations of these words would form books differing 
from one another still more widely. Now if he made yet another examination without the 
aid of a microscope, he would not find any other kind of elements that were more similar 
to one another than these letters, from a combination of which in different ways the letters 
themselves could be produced. But if he took a microscope, then indeed would he see 
the mode of formation of the letters from the perfectly homogeneous round points, by 
the different position & distribution of which the letters were depicted. 

99. This seems to me to be a sort of picture of what we perceive in Nature. Those Application of the 

i,- 7 - 7 . ,..,. r ., i T n i 1-111 illustration to the 

books, so many m number & so different in character are bodies, & those which belong analysis of Nature. 

to the different kingdoms are written as it were in different tongues. Of all of these, 

chemical analysis finds out certain fundamental constituents that are less unlike one another 

than the books ; these are the words. Yet these constituent substances have some sort 

of difference amongst themselves, & thus chemical analysis produces a large number of 

species of oils, earths & salts from different bodies. Further analysis of these, like that 

of the words, would disclose the letters that are still less unlike one another ; & finally, 

according to my Theory, the little homogeneous points would be obtained. These, just 

as the little black circles formed the letters, would form the diverse particles of diverse 

bodies through diverse arrangement alone. So far then the analogy derived from such a 


derivata non ad difformitatem, sed ad conformitatem elementorum nos ducit. 

Transitus a pro- ioo. Atque hoc demum pacto ex principiis certis & vulgo receptis, per legitimam, 

ad consectariorum seriem devenimus ad omnem illam, quam initio proposui, Theoriam, 
nimirum ad legem virium mutuarum, & ad constitutionem primorum materiae elementorum 
ex ilia ipsa virium lege derivatorum. [46] Videndum jam superest, quam uberes inde 
fructus per universam late Physicam colligantur, explicatis per earn unam praecipuis cor- 
porum proprietatibus, & Naturae phaenomenis. Sed antequam id aggredior, praecipuas 
quasdam e difficultatibus, quae contra Theoriam ipsam vel objectae jam sunt, vel in oculos 
etiam sponte incurrunt, dissolvam, uti promisi. 

Legem virium non ioi. Contra vires mutuas illud sclent objicere, illas esse occultas quasdam qualitates, 

in distans, a nec esse ve ^ etiam actionem in distans inducere. His satisfit notione virium exhibita numero 8, 
occuitam quaiita- & 9. Illud unum praeterea hie addo, admodum manifestas eas esse, quarum idea admodum 
facile efformatur, quarum existentia positive argumento evincitur, quarum effectus multi- 
plices continue oculis observantur. Sunt autem ejusmodi hae vires. Determinationis 
ad accessum, vel recessum idea efformatur admodum facile. Constat omnibus, quid sit 
accedere, quid recedere ; constat, quid sit esse indifferens, quid determinatum ; adeoque 
& determinationis ad accessum, vel recessum habetur idea admodum sane distincta. 
Argumenta itidem positiva, quae ipsius ejusmodi determinationis existentiam probant, 
superius prolata sunt. Demum etiam motus varii, qui ab ejusmodi viribus oriuntur, ut 
ubi corpus quoddam incurrit in aliud corpus, ubi partem solidi arreptam pars alia sequitur, 
ubi vaporum, vel elastrorum particulae se invicem repellunt, ubi gravia descendunt, hi 
motus, inquam, quotidie incurrant in oculos. Patet itidem saltern in genere forma curvae 
ejusmodi vires exprimentis. Haec omnia non occuitam, sed patentem reddunt ejusmodi 
virium legem. 

Quid adhuc lateat : IO2. Sunt quidem adhuc quaedam, quae ad earn pertinent, prorsus incognita, uti est 

admittendam om- numerus, & distantia intersectionum curvae cum axe, forma arcuum intermediorum, atque 

nino : quo pacto .. . ,. -11 i -11 i i i 

evitetur hie actio alia ejusmodi, quae quidem longe superant humanum captum, & quas me solus habuit 
in distans. omnia simul prae oculis, qui Mundum condidit ; sed id omnino nil officit. Nee sane 

id ipsum in causa esse debet, ut non admittatur illud, cujus existentiam novimus, & cujus 
proprietates plures, & effectus deprehendimus ; licet alia multa nobis incognita eodem 
pertinentia supersint. Sic aurum incognitam, occultamque substantiam nemo appellant, 
& multo minus ejusdem existentiam negabit idcirco, quod admodum probabile sit, plures 
alias latere ipsius proprietates, olim forte detegendas, uti'aliae tarn multae subinde detectae 
sunt, & quia non patet oculis, qui sit particularum ipsum componentium textus, quid, & 
qua ratione Natura ad ejus compositionem adhibeat. Quod autem pertinet ad actionem 
in distans, id abunde ibidem praevenimus, cum inde pateat fieri posse, ut punctum quodvis 
in se ipsum agat, & ad actionis directionem, ac energiam determinetur ab altero puncto, 
vel ut Deus juxta liberam sibi legem a se in Natura condenda stabilitam motum progignat 
in utroque pun-[47]-cto. Illud sane mihi est evidens, nihilo magis occuitam esse, vel explicatu, 
& captu difficilem productionem. motus per hasce vires pendentes a certis distantiis, quam 
sit productio motus vulgo concepta per immediatum impulsum, ubi ad motum determinat 
impenetrabilitas, quae itidem vel a corporum natura, vel a libera conditoris lege repeti 

sine impuisione 103. Et quidem hoc potius pacto, quam per impulsionem, in motuum causas, & leges 

Mst'hucus^'u^N inquirendum esse, illud etiam satis indicat, quod ubi hue usque, impuisione omissa, vires 

turam, & menus ex- adhibitae sunt a distantiis pendentes, ibi sane tantummodo accurate definita sunt omnia, 

phcajidam. impost- at q ue determinata, & ad calculum redacta cum phaenomenis congruunt ultra, quam sperare 

liceret, accuratissime. Ego quidem ejusmodi in explicando, ac determinando felicitatem 

nusquam alibi video in universa Physica, nisi tantummodo in Astronomia mechanica, quae 

abjectis vorticibus, atque omni impuisione submota, per gravitatem generalem absolvit 

omnia, ac in Theoria luminis, & colorum, in quibus per vires in aliqua distantia agentes, 

& reflexionem, & refractionem, & diffractionem Newtonus exposuit, ac priorum duarum, 

potissimum leges omnes per calculum, & Geometriam determinavit, & ubi ilia etiam, quae 

ad diversas vices facilioris transmissus, & facilioris reflexionis, quas Physici passim relinquunt 


consideration of Nature leads us not to non-uniformity but to uniformity of the 

100. Thus at length, from known principles that are commonly accepted, by a Pa^g ,, fro the 

... , , , .' . . r r , i i % i n-<i i T i proof of the Theory 

legitimate series of deductions, we have arrived at the whole of the I heory that I enunciated to the considera- 
at the start ; that is to say. at a law of mutual forces & the constitution of the primary tion f . objections 

, ' i i ,- i re XT i r i ' against it. 

elements of matter derived from that law of forces. Now it remains to be seen what a 
bountiful harvest is to be gathered throughout the wide field of general physics ; for from 
this one theory we obtain explanations of all the chief properties of bodies, & of the 
phenomena of Nature. But before I go on to that, I will give solutions of a few of the 
principal difficulties that have been raised against the Theory itself, as well as some that 
naturally meet the eye, according to the promise I made. 

10 1. The objection is frequently brought forward against mutual forces that they The law o{ forces 

, J . * . . , . . j. mi does not necessi- 

are some sort of mysterious qualities or that they necessitate action at a distance. This tate action at a 

is answered by the idea of forces outlined in Art. 8, & 9. In addition, I will make just distance, nor is it 

one remark, namely, that it is quite evident that these forces exist, that an idea of them quTuty. " 

can be easily formed, that their existence is demonstrated by direct reasoning, & that 

the manifold results that arise from them are a matter of continual ocular observation. 

Moreover these forces are of the following nature. The idea of a propensity to approach 

or of a propensity to recede is easily formed. For everybody knows what approach means, 

and what recession is ; everybody knows what it means to be indifferent, & what having 

a propensity means ; & thus the idea of a propensity to approach, or to recede, is perfectly 

distinctly obtained. Direct arguments, that prove the existence of this kind of propensity, 

have been given above. Lastly also, the various motions that arise from forces of this 

kind, such as when one body collides with another body, when one part of a solid is seized 

& another part follows it, when the particles of gases, & of springs, repel one another, 

when heavy bodies descend, these motions, I say, are of everyday occurrence before our 

eyes. It is evident also, at least in a general way, that the form of the curve represents 

forces of this kind. In all of these there is nothing mysterious ; on the contrary they all 

tend to make the law of forces of this kind perfectly plain. 

102. There are indeed certain things that relate to the law of forces of which we are What is so far un 
altogether ignorant, such as the number & distances of the intersections of the curve J^ ^ idmittedm 
with the axis, the shape of the intervening arcs, & other things of that sort ; these indeed ail detail ; the way 
far surpass human understanding, & He alone, Who founded the universe, had the whole ^ action 1 at h a to* 
before His eyes. But truly there is no reason on that account, why a thing, whose existence tance is eliminated, 
we fully recognize, & many of the properties & results of which are readily understood, 

should not be accepted ; although certainly there do remain many other things pertaining 
to it that are unknown to us. For instance, nobody would call gold an unknown & 
mysterious substance, & still less would deny its existence, simply because it is quite 
probable that many of its properties are unknown to us, to be discovered perhaps in the 
future, as so many others have been already discovered from time to time, or because it 
is not visually apparent what is the texture of the particles composing it, or why & in 
what way Nature adopts that particular composition. Again, as regards action at a distance, 
we amply guard against this by the same means ; for, if this is admitted, then it would 
be possible for any point to act upon itself, & to be determined as to its direction of action 
& energy apart from another point, or that God should produce in either point a motion 
according to some arbitrary law fixed by Him when founding the universe. To my mind 
indeed it is clear that motions produced by these forces depending on the distances are 
not a whit more mysterious, involved or difficult of understanding than the production 
of motion by immediate impulse as it is usually accepted ; in which impenetrability 
determines the motion, & the latter has to be derived just the same either from the nature 
of solid bodies, or from an arbitrary law of the founder of the universe. 

103. Now, that the investigation of the causes & laws of motion are better made As far as we have 
by my method, than through the idea of impulse, is sufficiently indicated by the fact that, f^' m0 re UI ciearJy 
where hitherto we have omitted impulse & employed forces depending on the distances, explained without 
only in this way has everything been accurately defined & determined, & when reduced g^ 1 what 

to calculation everything agrees with the phenomena with far more accuracy than we will be so too. 
could possibly have expected. Indeed I do not see anywhere such felicity in explaining 
& determining the matters of general physics, except only in celestial mechanics ; in 
which indeed, rejecting the idea of vortices, & doing away with that of impulse entirely, 
Newton gave a solution of everything by means of universal gravitation ; & in the theory 
of light & colours, where by means of forces acting at some distance he explained reflection, 
refraction & diffraction ; &, especially in the two first mentioned, he determined all 
the laws by calculus & Geometry. Here also those things depending on alternate fits 
of easier transmission & easier reflection, which physicists everywhere leave almost 

9 6 


fere intactas, ac alia multa admodum feliciter determinantur, explicanturque, quod & ego 
praestiti in dissertatione De Lumine, & praestabo hie in tertia parte ; cum in ceteris Physicae 
partibus plerumque explicationes habeantur subsidariis quibusdam principiis innixae & 
vagas admodum. Unde jam illud conjectare licet, si ab impulsione immediata penitus 
recedatur, & sibi constans ubique adhibeatur in Natura agendi ratio a distantiis pendens, 
multo sane facilius, & certius explicatum iri cetera ; quod quidem mihi omnino successit, 
ut patebit inferius, ubi Theoriam ipsam applicavero ad Naturam. 

Non fieri saltum in 

tracfiva ad repui- 

104. Solent & illud objicere, in hac potissimo Theoria virium committi saltum ilium, 
a( ^ quern evitandum ea inprimis admittitur ; fieri enim transitum ab attractionibus ad 
repulsiones per saltum, ubi nimirum a minima ultima repulsione ad minimam primam 
attractionem transitur. At isti continuitatis naturam, quam supra exposuimus, nequaquam 
intelligunt. Saltus, cui evitando Theoria inducitur, in eo consistit, quod ab una magnitudine 
ad aliam eatur sine transitu per intermedias. Id quidem non accidit in casu exposito. 
Assumatur quaecunque vis repulsiva utcunque parva ; turn quaecunque vis attractiva. 
Inter eas intercedunt omnes vires repulsivae minores usque ad zero, in quo habetur deter- 
minatio ad conservandum praecedentum statum quietis, vel motus uniformis in directum : 
turn omnes vires attractivae a z^-[48]-ro usque ad earn determinatam vim, & omnino nullus 
erit ex hisce omnibus intermediis statibus, quern aliquando non sint habitura puncta, quae 
a repulsione abeunt ad attractionem. Id ipsum facile erit contemplari in fig. i, in qua a 
vi repulsiva br ad attractionem dh itur utique continue motu puncti b ad d transeundo 
per omnes intermedias, & per ipsum zero in E sine ullo saltu ; cum ordinata in eo motu 
habitura sit omnes magnitudines minores priore br usque ad zero in E ; turn omnes oppositas 
majores usque ad posteriorem dh. Qui in ea veluti imagine mentis oculos defigat, is omnem 
apparentem difficultatem videbit plane sibi penitus evanescere. 

Nuiium esse post- JO P Q uo d autem additur de postremo repulsionis gradu, & primo attractionis nihil 

remum attractions, - 1 . ... r ,.... r , ., 

A: primum repuisio- sane probaret, quando etiam essent aliqui n gradus postrerm, & primi ; nam ab altero 
ms gradum, qm si eorum transiretur ad alterum per intermedium illud zero, & ex eo ipso, quod illi essent 

essent, adhuc tran- .....*. . ,. . . 

sire per omnes in- postremus, ac primus, mhil omitteretur mtermeaium, quae tamen sola intermedn omissio 
termedios. continuitatis legem evertit, & saltum inducit. Sed nee habetur ullus gradus postremus, 

aut primus, sicut nulla ibi est ordinata postrema, aut prima, nulla lineola omnium minima. 
Data quacunque lineola utcunque exigua, aliae ilia breviores habentur minores, ac minores 
ad infinitum sine ulla ultima, in quo ipso stat, uti supra etiam monuimus, continuitatis 
natura. Quamobrem qui primum, aut ultimum sibi confingit in lineola, in vi, in celeritatis 
gradu, in tempusculo, is naturam continuitatis ignorat, quam supra hie innui, & quam ego 
idcirco initio meae dissertationis De Lege Continuitatis abunde exposui. 

potest cuipiam saltern illud, ejusmodi legem virium, & curvam, quam in 
curvae, & duobus fig. I protuli, esse nimium complicatam, compositam, & irregularem, quae nimirum coalescat 
virium genenbus. ex } n g en ti numero arcuum jam attractivorum, jam repulsivorum, qui inter se nullo pacto 
cohaereant ; rem eo redire, ubi erat olim, cum apud Peripateticos pro singulis proprietatibus 
corporum singulae qualitates distinctae, & pro diversis speciebus diversae formae substantiales 
confingebaritur ad arbitrium. Sunt autem, qui & illud addant, repulsionem, & attractionem 
esse virium genera inter se diversa ; satius esse, alteram tantummodo adhibere, & repulsionem 
explicare tantummodo per attractionem minorem. 

repuisivam positive I0 7- Inprimis quod ad hoc postremum pertinet, satis patet, per positivam meae 

demonstrari prater Theoriae probationem immediate evinci repulsionem ita, ut a minore attractione repeti 
omnino non possit ; nam duae materiae particulae si etiam solae in Mundo essent, & ad se 
invicem cum aliqua velocitatum inaequalitate accederent, deberent utique ante contactum 
ad sequalitatem devenire vi, quse a nulla attractione pendere posset. 


untouched, & many other matters were most felicitously determined & explained by 
him ; & also that which I enunciated in the dissertation De Lumine, & will repeat in 
the third part of this work. For in other parts of physics most of the explanations are 
independent of, & disconnected from, one another, being based on several subsidiary 
principles. Hence we may now conclude that if, relinquishing all idea of immediate 
impulses, we employ a reason for the action of Nature that is everywhere the same & 
depends on the distances, the remainder will be explained with far greater ease & certainty ; 
& indeed it is altogether successful in my hands, as will be evident later, when I come 
to apply the Theory to Nature. 

104. It is very frequently objected that, in this Theory more especially, a sudden change There is no sudden 
is made in the forces, whilst the theory is to be accepted for the very purpose of avoiding sitfwPfrom aiTat- 
such a thing. For it is said that the transition from attractions to repulsions is made tractive to a repui- 
suddenly, namely, when we pass from the last extremely minute repulsive force to the 
first extremely minute attractive force. But those who raise these objections in no wise 
understand the nature of continuity, as it has been explained above. The sudden change, 
to avoid which the Theory has been brought forward, consists in the fact that a passage 
is made from one magnitude to another without going through the intermediate stages. 
Now this kind of thing does not take place in the case under consideration. Take any 
repulsive force, however small, & then any attractive force. Between these two there 
lie all the repulsive forces that are less than the former right down to zero, in which there 
is the propensity for preserving the original state of rest or of uniform motion in a straight 
line ; & also all the attractive forces from zero up to the prescribed attractive force, 
& there will be absolutely no one of all these intermediate states, which will not be possessed 
at some time or other by the points as they pass from repulsion to attraction. This can 
be readily understood from a study of Fig. I, where indeed the passage is made from the 
repulsive force br to the attractive force dh by the continuous motion of a point from b to 
d ; the passage is made through every intermediate stage, & through zero at E, without 
any sudden change. For in this motion there will be obtained as ordinates all magnitudes, 
less than the first one br, down to zero at E, & after that all magnitudes of opposite sign 
greater than zero as far as the last ordinate dh. Anyone, who will fix his intellectual vision 
on this as on a sort of pictorial illustration cannot fail to perceive for himself that all the 
apparent difficulty vanishes completely. 

i OS. Further, as regards what is said in addition about the last stage of repulsion & T^ 1 " 6 & no ! ast 

, r . . 11 11 TI 1111 stage of attraction, 

the first stage of attraction, it would really not matter, even if there were these so called an d no first for re- 
last & first stages ; for, from one of them to the other the passage would be made through puisjon ; and even 

6 ,.' . . , i c if there were, the 

the one intermediate stage, namely zero ; since it passes zero, & because they are the nrst passage would be 

& last, therefore no intermediate stage is omitted. Nevertheless the omission of this p ade through ail 

intermediate alone would upset the law of continuity, & introduce a sudden change. 

But, as a matter of fact, there cannot possibly be a last stage or a first ; just as there cannot 

be a last ordinate or a first in the curve, that is to say, a short line that is the least of 

them all. Given any short line, no matter how short, there will be others shorter than 

it, less & less in infinite succession without any limit whatever ; & in this, as we remarked 

also above, there lies the nature of continuity. Hence anyone who brings forward the 

idea of a first or a last in the case of a line, or a force, or a degree of velocity, or an 

interval of time, must be ignorant of continuity ; this I have mentioned before in this 

work, & also for this very reason I explained it very fully at the beginning of my 

dissertation De Lege Continuitatis. 

1 06. It may seem to some that at least a law of forces of this nature, & the curve g b ^ c s t t 10 ? he r appar d 
expressing it, which I gave in Fig. I, is very complicated, composite & irregular, being e nt composite cha- 
indeed made up of an immense number of arcs that are alternately attractive & repulsive, ^ te [ h f t 

& that these are joined together according to no definite plan ; & that it reduces to O f forces, 
the same thing as obtained amongst the ancients, since with the Peripatetics separate 
distinct qualities were invented for the several properties of bodies, & different substantial 
forms for different species. Moreover there are some who add that repulsion & attraction 
are kinds of forces that differ from one another ; & that it would be quite enough to 
use only the latter, & to explain repulsion merely as a smaller attraction. 

107. First of all, as regards the last objection, it is clear enough from what has been p^ssibie^o prove 
directly proved in my Theory that the existence of repulsion has been rigorously demonstrated directly the exist- 
in such a way that it cannot possibly be derived from the idea of a smaller attraction. For f^ce f apart PU from 
two particles of matter, if they were also the only particles in the universe, & approached attraction. 

one another with some difference of velocity, would be bound to attain to an equality of 
velocity on account of a force which could not possibly be derived from an attraction of 

any kind, 



tiva, & negativa. 

Hinc nihu pbstare, Io g i Deinde vef o quod pertinet ad duas diversas species attractionis, & repulsionis ; 

si diversi suit gene- ., . , ,. . ,^. r .r-ii-i i 

ris; sed esse ejus- id quidem licet ita se haberet, m-[49j-hil sane obesset, cum positive argumento evmcatur 
dem uti sunt posi- & re pulsio. & attractio, uti vidimus; at id ipsum est omnino falsum. Utraque vis ad 

. f . . . . ^ . . . 

eandem pertinet speciem, cum altera respectu alterms negativa sit, & negativa a positivis 
specie non differant. Alteram negativam esse respectu alterius, patet inde, quod tantum- 
modo differant in directione, quae in altera est prorsus opposita direction! alterius ; in 
altera enim habetur determinatio ad accessum, in altera ad recessum, & uti recessus, & 
accessus sunt positivum, ac negativum ; ita sunt pariter & determinationes ad ipsos. Quod 
autem negativum, & positivum ad eandem pertineant speciem, id sane patet vel ex eo 
principio : magis, W minus non differunt specie. Nam a positive per continuam subtrac- 
tionem, nimirum diminutionem, habentur prius minora positiva, turn zero, ac demum 
negativa, continuando subtractionem eandem. 

Probatio hujus a 
progressu, & re- 
gressu, in fluvio. 

109. Id facile patet exemplis solitis. Eat aliquis contra fluvii directionem versus locum 
aliquem superiori alveo proximum, & singulis minutis perficiat remis, vel vento too hexapedas, 
dum a cursu fluvii retroagitur per hexapedas 40 ; is habet progressum hexapedarum 60 
singulis minutis. Crescat autem continue impetus fluvii ita, ut retroagatur per 50, turn per 
60, 70, 80, 90, 100, no, 120, &c. Is progredietur per 50, 40, 30, 20, 10, nihil ; turn 
regredietur per 10, 20, quae erunt negativa priorum ; nam erat prius 100 50, 100 60, 
10070,100 80,100 90, turn 100 100=0,100 no, = 10, 100 120 = 20, et ita 
porro. Continua imminutione, sive subtractione itum est a positivis in negativa, a 
progressu ad regressum, in quibus idcirco eadem species mansit, non duae diversae. 

Probatio ex Alge- 
bra, & Geometria : 
applicatio ad omnes 
quantitates varia- 

An habeatur trans- 
itus e positivis in 
negativa ; investi- 
gatio ex sola curv- 
arum natura. 




FIG. ii. 

i to. Idem autem & algebraicis formulis, & geometricis lineis satis manifeste ostenditur. 
Sit formula 10 x, & pro x ponantur valores 6, 7, 8, 9, 10, n, 12, &c. ; valor formulae 
exhibebit 4, 3, 2, I, o, I, 2, &c., quod eodem redit, ubi erat superius in progressu, & 
regressu, qui exprimerentur simulper formulam 10 x. Eadem ilia formula per continuam 
mutationem valoris x migrat e valore positive in negativum, qui aeque ad eandem formulam 
pertinent. Eodem pacto in Geometria in fig. 
u,siduae lineae MN, OP referantur invicem 
per ordinatas AB, CD, &c. parallelas inter se, 
secent autem se in E ; continue motu ipsius 
ordinatae a positive abitur in negativum, mutata 
directione AB, CD, quae hie habentur pro 
positivis, in FG, HI, post evanescentiam in E. 
Ad eandem lineam continuam OEP aeque 
pertinet omnis ea ordinatarum series, nee est 
altera linea, alter locus geometricus OE, ubi 
ordinatae sunt positivae, ac EP, ubi sunt nega- 
tivae. Jam vero variabilis quantitatis cujusvis 
natura, & lex plerumque per formulam aliquam analyticam, semper per ordinatas ad lineam 
aliquam exprimi potest ; si [50] enim singulis ejus statibus ducatur perpendicularis 
respondens ; vertices omnium ejusmodi perpendicularium erunt utique ad lineam quandam 
continuam. Si ea linea nusquam ad alteram abeat axis partem, si ea formula nullum valorem 
negativum habeat ; ilia etiam quantitas semper positiva manebit. Sed si mutet latus linea, 
vel formula valoris signum ; ipsa ilia quantitatis debebit itidem ejusmodi mutationem 
habere. Ut autem a formulae, vel lineae exprimentis natura, & positione respectu axis 
mutatio pendet ; ita mutatio eadem a natura quantitatis illius pendebit ; & ut nee duas 
formulae, nee duae lineae speciei diversae sunt, quae positiva exhibent, & negativa ; ita nee in 
ea quantitate duae erunt naturae, duae species, quarum altera exhibeat positiva, altera 
negativa, ut altera progressus, altera regressus ; altera accessus, altera recessus ; & hie altera 
attractiones, altera repulsiones exhibeat ; sed eadem erit, unica, & ad eandem pertinens 
quantitatis speciem tota. 

in. Quin immo hie locum habet argumentum quoddam, quo usus sum in dissertatione 
De Lege Continuitatis, quo nimirum Theoria virium attractivarum, & repulsivarum pro 
diversis distantiis, multo magis rationi consentanea evincitur, quam Theoria ^ virium 
tantummodo attractivarum, vel tantummodo repulsivarum. Fingamus illud, nos ignorare 
penitus, quodnam virium genus in Natura existat, an tantummodo attractivarum, vel 
repulsivarum tantummodo, an utrumque simul : hac sane ratiocinatione ad earn perquisi- 
tionem uti liceret. Erit utique aliqua linea continua, quae per suas ordinatas ad axem 
exprimentem distantias, vires ipsas determinabit, & prout ipsa axem secuerit, vel non 



108. Next, as regards attraction & repulsion being of different species, even if it Hence it does not 
were a fact that they were so, it would not matter in the slightest degree, since by rigorous S at different h kmds! 
argument the existence of both attraction & repulsion is proved, as we have seen ; but but as a matter of 
really the supposition is untrue. Both kinds of force belong to the same species ; for one same^kmdnusVas 
is negative with regard to the other, & a negative does not differ in species from positives. a positive and a 
That the one is negative with regard to the other is evident from the fact that they only negatlve are so - 
differ in direction, the direction of one being exactly the opposite of the direction of the 

other ; for in the one there is a propensity to approach, in the other a propensity to recede ; 
& just as approach & recession are positive & negative, so also are the propensities 
for these equally so. Further, that such a negative & a positive belong to the same species, 
is quite evident from the principle the greater & the less are not different in kind. For 
from a positive by continual subtraction, or diminution, we first obtain less positives, then 
zero, & finally negatives, the same subtraction being continued throughout. 

109. The matter is easily made clear by the usual illustrations. Suppose a man Demonstration by 
to go against the current of a river to some place on the bank up-stream; & suppose "veTndretrogS 
that he succeeds in doing, either by rowing or sailing, 100 fathoms a minute, whilst he motion on a river. 
is carried back by the current of the river through 40 fathoms ; then he will get forward 

a distance of 60 fathoms a minute. Now suppose that the strength of the current continually 
increases in such a way that he is carried back first 50, then 60, 70, 80, 90, ipo, no, 120, 
&c. fathoms per minute. His forward motion will be successively 50, 40, 30, 20, 10 fathoms 
per minute, then nothing, & then he will be carried backward through 10, 20, &c. fathoms 
a minute ; & these latter motions are the negatives of the former. For first of all we 
had 100 50, 100 60, 100 70, 100 80, 100 90, then 100 100 (which = o), 
then 100 no (which = 10), 100 120 (which = 20), and so on. By a continual 
diminution or subtraction we have passed from positives to negatives, from a progressive 
to a retrograde motion ; & therefore in these there was a continuance of the same species, 
and there were not two different species. 

no. Further, the same thing is shown plainly enough by algebraical formulae, & Proof from algebra 
by lines in geometry. Consider the formula 10 x, & for x substitute the values, 6, pucatfon^ O y : a n 
7, 8, 9, 10, n, 12, &c. ; then the value of the formula will give in succession 4, 3, 2, variable quantities. 
I, o, i, 2, &c. ; & this comes to the same thing as we had above in the case of the 
progressive & retrograde motion, which may be expressed by the formula 10 x, all 
together. This same formula passes, by a continuous change in the value of x, from a 
positive value to a negative, which equally belong to the same formula. In the same 
manner in geometry, in Fig. 1 1, if two lines MN, OP are referred to one another by ordinates 
AB, CD, & also cut one another in E ; then by a continuous motion of the ordinate 
itself it passes from positive to negative, the direction of AB, CD, which are here taken 
to be positive, being changed to that of FG, HI, after evanescence at E. To the same 
continuous line OEP belongs equally the whole of this series of ordinates ; & OE, where 
the ordinates are positive, is not a different line, or geometrical locus from EP, where the 
ordinates are negative. Now the nature of any variable quantity, & very frequently 
also the law, can be expressed by an algebraical formula, & can always be expressed by 
some line ; for if a perpendicular be drawn to correspond to each separate state of the 
quantity, the vertices of all perpendiculars so drawn will undoubtedly form some continuous 
line. If the line never passes over to the other side of the axis, if the formula has no negative 
value, then also the quantity will always remain positive. But if the line changes side, 
or the formula the sign of its value, then the quantity itself must also have a change of the 
same kind. Further, as the change depends on the nature of the formula & the line 
expressing it, & its position with respect to the axis ; so also the same change will depend 
on the nature of the quantity ; & just as there are not two formulas, or two lines of 
different species to represent the positives & the negatives, so also there will not be in the 
quantity two natures, or two species, of which the one yields positives & the other negatives, 
as the one a progressive & the other a retrograde motion, the one approach & the other 
recession, & in the matter under consideration the one will give attractions & the other 
repulsions. But it will be one & the same nature & wholly belonging to the same 
spec es of quantity. 

in. Lastly, this is the proper place for me to bring forward an argument that I used whether there can 

i i . T\ T /-. -T. r . ,..,!.. , .be a transition 

in the dissertation De Lege Continmtatis ; by it indeed it is proved that a theory of attractive {rom positive to 
& repulsive forces for different distances is far more reasonable than one of attractive negative ; mves- 
forces only, or of repulsive forces only. Let us imagine that we are quite ignorant of the of 8 thenature of the 
kind of forces that exist in Nature, whether they are only attractive or only repulsive, or curv e only, 
both ; it would be allowable to use the following reasoning to help us to investigate the 
matter. Without doubt there will be some continuous line which, by means of ordinates 
drawn from it to an axis representing distances, will determine the forces ; & according 



cent. qi 

secuerit ; vires erunt alibi attractive, alibi repulsivae ; vel ubique attractive tantum, aut 
repulsive tantum. Videndum igitur, an sit ration! consentaneum magis, lineam ejus 
naturae, & positionis censere, ut axem alicubi secet, an ut non secet. 

Transitum deduci U2. Inter rectas axem rectilineum unica parallela ducta per quod vis datum punctum 

sint 0> curvse, P quas non secat j omnes alie numero infinitae secant alicubi. Curvarum nulla est, quam infinitae 
recte secent, quam numero rectae secare non possint ; & licet aliquae curvae ejus naturae sint, ut eas aliquae rectae 
non secent ; tamen & eas ipsas aliae infinite numero recte secant, & infinite numero curve, 
quod Geometrie sublimioris peritis est notissimum, sunt ejus nature, ut nulla prorsus sit 
recta linea, a qua possint non secari. Hujusmodi ex. gr. est parabola ilia, cujus ordinate 
sunt in ratione triplicata abscissarum. Quare infinite numero curve sunt, & infinite 
numero rectae, que sectionem necessario habeant, pro quavis recta, que non habeat, & nulla 
est curva, que sectionem cum axe habere non possit. Ergo inter casus possibles multo 
plures sunt ii, qui sectionem admittunt, quam qui ea careant ; adeoque seclusis rationibus 
aliis omnibus, & sola casuum probabilitate, & rei [51] natura abstracte considerata, multo 
magis rationi consentaneum est, censere lineam illam, que vires exprimat, esse unam ex iis, 
que axem secant, quam ex iis, que non secant, adeoque & ejusmodi esse virium legem, ut 
attractiones, & repulsiones exhibeat simul pro diversis distantiis, quam ut alteras tantummodo 
referat ; usque adeo rei natura considerata non solam attractionem, vel solam repulsionem, 
sed utramque nobis objicit simul. 


a recta. 


Ulterior perqui- u* ged eodem argumento licet ultenus quoque progredi, & primum etiam difficultatis 

sitio: curvarum J , o -j i 

genera : quo aiti- caput amovere, quod a sectionum, & idcirco etiam arcuum jam attractivorum, jam repulsi- 
ores, eo in piuribus vorum multiplicitate desumitur. Curvas lineas Geometre in quasdam classes dividunt 

uni , ......... 

P e anaiyseos, que earum naturam expnmit per mas, quas Analyste appellant, equationes, 
& que ad varies gradus ascendunt. Aequationes primi gradus exprimunt rectas ; equati- 
ones secundi gradus curvas primi generis ; equationes tertii gradus curvas secundi generis, 
atque ita porro ; & sunt curve, que omnes gradus transcendunt finite Algebre, & que 
idcirco dicuntur transcendentes. Porro illud demonstrant Geometre in Analysi ad 
Geometriam applicata, lineas, que exprimuntur per equationem primi gradus, posse 
secari a recta in unico puncto ; que equationem habent gradus secundi, tertii, & ita porro, 
secari posse a recta in punctis duobus, tribus, & ita porro : unde fit, ut curva noni, vel 
nonagesimi noni generis secari possit a recta in punctis decem, vel centum. 


sum plures in eo- 

J am vero curvae primi generis sunt tantummodo tres conice sectiones, ellipis, 
parabola, hyperbola, adnumerato ellipsibus etiam circulo, que quidem veteribus quoque 
Geometris innotuerunt. Curvas secundi generis enumeravit Newtonus omnium primus, 
& sunt circiter octoginta ; curvarum generis tertii nemo adhuc numerum exhibuit accura- 
tum, & mirum sane, quantus sit is ipse illarum numerus. Sed quo altius assurgit curve 
genus, eo plures in eo genere sunt curve, progressione ita in immensum crescente, ut ubi 
aliquanto altius ascenderit genus ipsum, numerus curvarum omnem superet humane 
imaginationis vim. Idem nimirum ibi accidit, quod in combinationibus terminorum, de 
quibus supra mentionem fecimus, ubi diximus a 24 litterulis omnes exhiberi voces linguarum 
omnium, & que fuerunt, aut sunt, & que esse possunt. 

Deductio inde piu- jjr I n de iam pronum est argumentationem hujusmodi instituere. Numerus 

rimarum mtersec- .. J .. . ,.. J .. 

tionum, axis, & linearum, que axem secare possint in punctis quamplunmis, est in immensum major earum 
curvae exprimentis numero, quae non possint, nisi in paucis, vel unico : igitur ubi agitur de linea exprimente 
legem virium, ei, qui nihil aliunde sciat, in immensum probabilius erit, ejusmodi lineam 
esse ex prio-[52]-rum genere unam, quam ex genere posteriorum, adeoque ipsam virium 
naturam plurimos requirere transitus ab attractionibus ad repulsiones, & vice versa, quam 
paucos, vel nullum. 

- Sed omissa ista conjecturali argumentatione quadam, formam curve exprimentis 

simpiicem: in quo vires positive argumento a phenomenis Nature deducto nos supra determinavimus cum 
plurimis intersectionibus, que transitus ejusmodi quamplurimos exhibeant. Nee ejusmodi 
curva debet esse e piuribus arcubus temere compaginata, & compacta : diximus enim, 

11 * 


as it will cut the axis, or will not, the forces will be either partly attractive & partly 
repulsive, or everywhere only attractive or only repulsive. Accordingly it is to be seen 
if it is more reasonable to suppose that a line of this nature & position cuts the axis anywhere, 
or does not. 

112. Amongst straight lines there is only one, drawn parallel to the rectilinear axis, intersection is to 
through any given point that does not cut the axis; all the rest (infinite in number) will the factThat t f here 
cut it somewhere. There is no curve that an infinite number of straight lines cannot cut ; are more lines that 
& although there are some curves of such a nature that some straight lines do not cut them, thL^es^hat^o 
yet there are an infinite number of other straight lines that do cut these curves ; & there not. 

are an infinite number of curves, as is well-known to those versed in higher geometry, of 
such a nature that there is absolutely not a single straight line by which they cannot be 
cut. An example of this kind of curve is that parabola, in which the ordinates are in the 
triplicate ratio of the abscissae. Hence there are an infinite number of curves & an 
infinite number of straight lines which necessarily have intersection, corresponding to any 
straight line that has not ; & there is no curve that cannot have intersection with an 
axis. Therefore amongst the cases that are possible, there are far more curves that admit 
intersection than those that are free from it ; hence, putting all other reasons on one side, 
& considering only the probability of the cases & the nature of the matter on its own 
merits, it is far more reasonable to suppose that the line representing the forces is one of 
those, which cut the axis, than one of those that do not cut it. Thus the law of forces 
is such that it yields both attractions & repulsions (for different distances), rather than 
such that it deals with either alone. Thus far the nature of the matter has been considered, 
with the result that it presents to us, not attraction alone, nor repulsion alone, but both of 
these together. 

113. But we can also proceed still further adopting the same line of argument, & Further investiga- 
first of all remove the chief point of the difficulty, that is derived from the multiplicity S^L.^JILhi 

ri* */i i i p i i i curves , nit, iijgiicr 

of the intersections, & consequently also of the arcs alternately attractive & repulsive, their order, the 

Geometricians divide curves into certain classes by the help of analysis, which expresses wWcV^a ^teaight 

their nature by what the analysts call equations ; these equations rise to various degrees, line can cut them. 

Equations of the first degree represent straight lines, equations of the second degree represent 

curves of the first class, equations of the third degree curves of the second class, & so on. 

There are also curves which transcend all degrees of finite algebra, & on that account 

these are called transcendental curves. Further, geometricians prove, in analysis applied 

to geometry, that lines that are expressed by equations of the first degree can be cut by a 

straight line in one point only ; those that have equations of the second, third, & higher 

degrees can be cut by a straight line in two, three, & more points respectively. Hence 

it comes about that a curve of the ninth, or the ninety-ninth class can be cut by a straight 

line in ten, or in a hundred, points. 

114. Now there are only three curves of the first class, namely the conic sections, the As the class gets 
parabola, the ellipse & the hyperbola; the circle is included under the name of ellipse; gh " 

of that 
& these three curves were known to the ancient geometricians also. Newton was the class becomes im- 

first of all persons to enumerate the curves of the second class, & there are about eighty mensel y greater. 

of them. Nobody hitherto has stated an exact number for the curves of the third class ; 

& it is really wonderful how great is the number of these curves. Moreover, the higher 

the class of the curve becomes, the more curves there are in that class, according to a 

progression that increases in such immensity that, when the class has risen but a little higher, 

the number of curves will altogether surpass the fullest power of the human imagination. 

Indeed the same thing happens in this case as in combinations of terms ; we mentioned 

the latter above, when we said that by means of 24 little letters there can be 

expressed all the words of all languages that ever have been, or are, or can be in 

the future. 

115. From what has been said above we are led to set up the following line of argument. Hence we deduce 
The number of lines that can cut the axis in very many points is immensely greater than that there . are ^ 

, , , .... ' , ' r . . . ' f> many intersections 

the number of those that can cut it in a few points only, or in a single point. Hence, when O f the axis and the 
the line representing the law of forces is in question, it will appear to one. who otherwise ? urve representing 

i i i i i r i 111 , forces. 

knows nothing about its nature, that it is immensely more probable that the curve is of 
the first kind than that it is of the second kind ; & therefore that the nature of the forces 
must be such as requires a very large number of transitions from attractions to repulsions 
& back again, rather than a small number or none at all. 

116. But, omitting this somewhat conjectural line of reasoning, we have already it may be that the 
determined, by what has been said above, the form of the curve representing forces by a j|? I^SSlJ 8 

. ' rxr /iii simple , tnecuarac- 

ngorous argument derived trom the phenomena of Nature, & that there are very many teristic of simplicity 
intersections which represent just as many of these transitions. Further, a curve of this mcurves - 



notum esse Geometris, infinita esse curvarum genera, quae ex ipsa natura sua debeant axem 
in plurimis secare punctis, adeoque & circa ipsum sinuari ; sed praeter hanc generalem 
responsionem desumptam a generali curvarum natura, in dissertatione De Lege Firium in 
Natura existentium ego quidem directe demonstravi, curvam illius ipsius formae, cujusmodi 
ea est, quam in fig. i exhibui, simplicem esse posse, non ex arcubus diversarum curvarum 
compositam. Simplicem autem ejusmodi curvam affirmavi esse posse : earn enim simplicem 
appello, quae tota est uniformis naturae, quae in Analysi exponi possit per aequationem non 
resolubilem in plures, e quarum multiplicatione eadem componatur cujuscunque demum 
ea curva sit generis, quotcunque habeat flexus, & contorsiones. Nobis quidem altiorum 
generum curvae videntur minus simplices ; quh nimirum nostrae humanae menti, uti pluribus 
ostendi in dissertatione De Maris Aestu, & in Stayanis Supplementis, recta linea videtur 
omnium simplicissima, cujus congruentiam in superpositione intuemur mentis oculis 
evidentissime, & ex qua una omnem nos homines nostram derivamus Geometriam ; ac 
idcirco, quae lineae a recta recedunt magis, & discrepant, illas habemus pro compositis, & 
magis ab ea simplicitate, quam nobis confinximus, recedentibus. At vero lineae continuae, 
& uniformis naturae omnes in se ipsis sunt aeque simplices ; & aliud mentium genus, quod 
cujuspiam ex ipsis proprietatem aliquam aeque evidenter intueretur, ac nos intuemur 
congruentiam rectarum, illas maxime simplices esse crederet curvas lineas, ex ilia earum 
proprietate longe alterius Geometrise sibi elementa conficeret, & ad illam ceteras referret 
lineas, ut nos ad rectam referimus ; quas quidem mentes si aliquam ex. gr. parabolae pro- 
prietatem intime perspicerent, atque intuerentur, non illud quaarerent, quod nostri 
Geometrae quaerunt, ut parabolam rectificarent, sed, si ita loqui fas est, ut rectam 

Problema continens 1 1 7. Et quidem analyseos ipsius profundiorem cognitionem requirit ipsa investigatio 

naturam curvaeana- aequationis, qua possit exprimi curva ems formae, quae meam exhibet virium legem. 

lytice expnmendam. /! j- 11 ji -i 

Quamobrem hie tantummodo exponam conditiones, quas ipsa curva habere debet, & quibus 
aequatio ibi inventa satis facere [53] debeat. (c) Continetur autem id ipsum num. 75, 
illius dissertationis, ubi habetur hujusmodi Problema : Invenire naturam curvce, cujus 
abscissis exprimentibus distantias, ordinal exprimant vires, mutatis distantiis utcunque 
mutatas, y in datis quotcunque limitibus transeuntes e repulsivis in attractivas, ac ex attractivis 
in repulsivas, in minimis autem distantiis repulsivas, W ita crescentes, ut sint pares extinguendce 
cuicunque velocitati utcunque magnce. Proposito problemate illud addo : quoniam posuimus 
mutatis distantiis utcunque mutatas, complectitur propositio etiam rationem quee ad rationem 
reciprocam duplicatam distantiarum accedat, quantum libuerit, in quibusdam satis magnis 

Conditiones ejus 

1 18. His propositis numero illo 75, sequenti numero propono sequentes sex conditiones, 
quae requirantur, & sufficiant ad habendam curvam, quse quaeritur. Primo : ut sit regularis, 
ac simplex, & non composita ex aggregate arcuum diversarum curvarum. Secundo : ut secet 
axem C'AC figures i. tantum in punctis quibusdam datis ad binas distantias AE', AE ; AG', 
AG ; y ita porro cequales (d) bine, y inde. Tertio : ut singulis abscissis respondeant singulcs 
ordinatcf. ( e ) Quarto : ut sumptis abscissis cequalibus hinc, y inde ab A, respondeant ordinal* 

(c) Qui velit ipsam rei determinationem videre, poterit hie in fine, ubi Supphmentorum, 3. exhibebitur solutio 
problematis, qua in memorata dissertatione continetur a num. 77. ad no. Sed W numerorum ordo, & figurarum 
mutabitur, ut cum reliquis hujusce operis cohtereat. 

Addetur prieterea eidem . postremum scholium pertinens ad qu<sstionem agitatam ante has aliquot annos Parisiis ; 
an vis mutua inter materite particulas debeat omnino exprimi per solam aliquam distantiee potenttam, an possit per 
aliquam ejus functionem ; W constabit, posse utique per junctionem, ut hie ego presto, qute uti superiore numero de curvts 
est dictum, est in se eeque simplex etiam, ubi nobis potentias ad ejus expressionem adhibentibus videatur admodum 

(d) Id, ut y quarta conditio, requiritur, ut curva utrinque sit sui similis, quod ipsam magis uniformem reddit ; 
quanquam de illo crure, quod est citra asymptotum AB, nihil est, quod soliciti simus ; cum ob vim repulsivam imminutis 
distantiis ita in infinitum excrescentem, non possit abscissa distantiam exprimens unquam evadere zero, W abire in 

(e) Nam singulis distantiis singulte vires respondent. 


kind is not bound to be built up by connecting together a number of independent arcs. 
For, as I said, it is well known to Geometricians that there are an infinite number of classes 
of curves that, from their very nature, must cut the axis in a very large number of points, 
& therefore also wind themselves about it. Moreover, in addition to this general answer 
to the objector, derived from the general nature of curves, in my dissertation De Lege 
Firium in Natura existentium, I indeed proved in a straightforward manner that a curve, 
of the form that I have given in Fig. i, might be simple & not built up of arcs of several 
different curves. Further, I asserted that a simple curve of this kind was perfectly feasible ; 
for I call a curve simple, when the whole of it is of one uniform nature. In analysis, this 
can be expressed by an equation that is not capable of being resolved into several other 
equations, such that the former is formed from the latter by multiplication ; & that too, 
no matter of what class the curve may be, or how many flexures or windings it may have. 
It is true that the curves of higher classes seem to us to be less simple ; this is so because, 
as I have shown in several places in the dissertation De Marts Aestu, & the supplements 
to Stay's Philosophy, a straight line seems to our human mind to be the simplest of all 
lines ; for we get a real clear mental perception of the congruence on superposition in the 
case of a straight line, & from this we human beings form the whole of our geometry. 
On this account, the more that lines depart from straightness & the more they differ, 
the more we consider them to be composite & to depart from that simplicity that we have 
set up as our standard. But really all lines that are continuous & of uniform nature 
are just as simple as one another. Another kind of mind, which might form an equally 
clear mental perception of some property of any one of these curves, as we do the congruence 
of straight lines, might believe these curves to be the simplest of all & from that property 
of these curves build up the elements of a far different geometry, referring all other curves 
to that one, just as we compare them with a straight line. Indeed, these minds, if they 
noticed & formed an extremely clear perception of some property of, say, the parabola, 
would not seek, as our geometricians do, to rectify the parabola ; they would endeavour, 
if one may use the words, to parabolify a straight line. 

1 17. The investigation of the equation, by which a curve of the form that will represent p T bl !J n . 

' i ~ j' i i j f i ir 1T71. r "* the analytical 

my law of forces can be expressed, requires a deeper knowledge 01 analysis itselt. Wnereiore expression of the 

I will here do no more than set out the necessary requirements that the curve must fulfil nature of the curve. 

& those that the equation thereby discovered must satisfy. (c) It is the subject of Art. 75 

of the dissertation De Lege Firium, where the following problem is proposed. Required 

to find the nature of the curve, whose abscissa represent distances & whose ordinates represent 

forces that are changed as the distances are changed in any manner, y pass from attractive 

forces to repulsive, & from repulsive to attractive, at any given number of limit-points ; further, 

the forces are repulsive at extremely small distances and increase in such a manner that they 

are capable of destroying any velocity, however great it may be. To the problem as there 

proposed I now add the following : As we have used the words are changed as the distances 

are changed in any manner, the proposition includes also the ratio that approaches as nearly 

as you please to the reciprocal ratio of the squares of the distances, whenever the distances are 

sufficiently great. 

1 1 8. In addition to what is proposed in this Art. 75, I set forth in the article that The * of 
follows it the following six conditions ; these are the necessary and sufficient conditions 

for determining the curve that is required. 

(i) The curve is regular & simple, & not compounded of a number of arcs of different curves. 

(ii) It shall cut the axis C'AC of Fig. I, only in certain given points, whose distances, 
AE',AE, AG', AG, and so on, are equal (<t) in pairs on each side of A [see p. 80]. 

(iii) To each abscissa there shall correspond one ordinate y one only, (f) 

(iv) To equal abscisses, taken one on each side of A, there shall correspond equal ordinates. 

(c) Anyone who desires to see the solution of the -problem will be able to do seat the end of this work; it will be 
found in 3 of the Supplements ; it is the solution of the problem, as it was given in the dissertation mentioned above, 
from Art. 77 to no. But here both the numbering of the articles W of the diagrams have been changed, so as to 
agree with the rest of the work. In addition, at the end of this section, there will be found a final note dealing 
with a question that was discussed some years ago in Paris. Namely, whether the mutual force between particles of mat- 
ter is bound to be expressible by some one power of the distance only, or by some function of the distance. It will be 
evident that at any rate it may be expressible by a function as I here assert ; y that function, as has been stated in the 
article above, is perfectly simple in itself also ; whereas, if we adhere to an expression by means of powers, the curve will 
seem to be altogether complex. 

(d) This, y the fourth condition too, is required to make the curve symmetrical, thus giving it greater uniformity ; 
although we are not concerned with the branch on the other side of the asymptote AB at all. For, on account of the 
repulsive force at very small distances increasing indefinitely in such a manner as postulated, it is impossible that the 
abscissa that represents the distance should ever become zero y then become negative. 

(e) For to each distance one force, & and only one, corresponds. 



czquales. Quinto : ut babeant rectam AB pro asymptoto, area asymptotica BAED existente ( ) 
infinita. Sexto : ut arcus binis quibuscunque intersectionibus terminati possint variari, ut 
libuerit, fcf? ad quascunque distantias recedere ab axe C'AC, ac accedcre ad quoscunque quarum- 
cunque curvarum arcus, quantum libuerit, eos secanda, vel tangendo, vel osculando ubicunque, 
ff quomodocunque libuerit. 

soiutio IrT U attrac IS4] IT 9- Verum quod ad multiplicitatem virium pertinet, quas diversis jam Physici 
tionem gravitatis nominibus appellant, illud hie etiam notari potest, si quis singulas seorsim considerare 
velit, licere illud etiam, hanc curvam in se unicam per resolutionem virium cogitatione 
nostra, atque fictione quadam, dividere in plures. Si ex. gr. quis velit considerare in materia 
gravitatem generalem accurate reciprocam distantiarum quadratis ; poterit sane is describere 
ex parte attractiva hyperbolam illam, quae habeat accurate ordinatas in ratione reciproca 
duplicata distantiarum, quse quidem erit quaedam velut continuatio cruris VTS, turn singulis 
ordinatis ag, dh curvae virium expressae in fig. I. adjungere ordinatas hujus novae hyperbolae 
ad partes AB incipiendo a punctis curvae g, b, & eo pacto orietur nova quaedam curva, quae 
versus partes pV coincidet ad sensum cum axe oC, in reliquis locis ab eo distabit, & contor- 
quebitur etiam circa ipsum, si vertices F, K, O distiterint ab axe magis, quam distet ibidem 
hyperbola ilia. Turn poterit dici, puncta omnia materiae habere gravitatem decrescentem 
accurate in ratione reciproca duplicata distantiarum, & simul habere vim aliam expressam 
ab ilia nova curva : nam idem erit, concipere simul hasce binas leges virium, ac illam 
praecedentem unicam, & iidem effectus orientur. 

Hujus posterioris 
vis resolutio in alias 

1 20. Eodem pacto haec nova curva potest dividi in alias duas, vel plures, concipiendo 
aliam quamcunque vim, ut ut accurate servantem quasdam determinatas leges, sed simul 
mutando curvam jam genitam, translatis ejus punctis per intervalla aequalia ordinatis 
respondentibus novae legi ass.umptae. Hoc pacto habebuntur plures etiam vires diversae, 
quod aliquando, ut in resolutione virium accidere diximus, inserviet ad faciliorem deter- 
minationem effectuum, & ea erit itidem vera virium resolutio quaedam ; sed id omne erit 
nostrae mentis partus quidam ; nam reipsa unica lex virium habebitur, quam in fig. I . 
exposui, & quae ex omnibus ejusmodi legibus componetur. 

Non obesse theo- 
r i a m gravitatis ; 

distantiis locum 
non habet. 

121. Quoniam autem hie mentio injecta est gravitatis decrescentis accurate in ratione 
cujusiex1n a minimis reciproca duplicata distantiarum ; cavendum, ne cui difficultatem aliquam pariat illud, 
'""" m quod apud Physicos, & potissimum apud Astronomiae mechanicae cultores, habetur pro 
comperto, gravitatem decrescere in ratione reciproca duplicata distantiarum accurate, 
cum in hac mea Theoria lex virium discedat plurimum ab ipsa ratione reciproca duplicata 
distantiarum. Inprimis in minoribus distantiis vis integra, quam in se mutuo exercent 
particulae, omnino plurimum discrepat a gravitate, quae sit in ratione reciproca duplicata 
distantiarum. Nam & vapores, qui tantam exercent vim ad se expandendos, repulsionem 
habent utique in illis minimis distantiis a se invicem, non attractionem ; & ipsa attractio, 
quae in cohaesione se prodit, est ilia quidem in immensum major, quam quae ex generali 
gravitate consequitur ; cum ex ipsis Newtoni compertis attractio gravitati respondens [55] 
in globes homogeneos diversarum diametrorum sit in eadem ratione, in qua sunt globorum 
diametri, adeoque vis ejusmodi in exiguam particulam est eo minor gravitate corporum in 
Terram, quo minor est diameter particulae diametro totius Terrae, adeoque penitus insen- 
sibilis. Et idcirco Newtonus aliam admisit vim pro cohaesione, quae decrescat in ratione 
majore, quam sit reciproca duplicata distantiarum ; & multi ex Newtonianis admiserunt 

vim respondentem huic formulae - 3 + - v cujus prior pars respectu posterioris sit in 

immensum minor, ubi x sit in immensum major unitate assumpta ; sit vero major, ubi x 
sit in immensum minor, ut idcirco in satis magnis distantiis evanescente ad sensum prima 
parte, vis remaneat quam proxime in ratione reciproca duplicata distantiarum x, in minimis 
vero distantiis sit quam proxime in ratione reciproca triplicata : usque adeo ne apud 
Newtonianos quidem servatur omnino accurate ratio duplicata distantiarum. 



I2 2. Demonstravit quidem Newtonus, in ellipsibus planetariis, earn, quam Astronomi 
q^ampro lineam apsidum nominant, & est axis ellipseos, habituram ingentem motum, si ratio virium 
ime, non accurate. a re ciproca duplicata distantiarum aliquanto magis aberret, cumque ad sensum quiescant 

(f) Id requiritur, quia in Mecbanica demonstrator, aream curves, cujus abscissa fxprimant distantias, 13 ordinatx 
vires, exprimere incrementum, vel decrementum quadrati velocitatis : quare ut illte vires sint pares extinguendte veloci- 
tati cuivis utcunque magna, debet ilia area esse omni finita major. 


(v) The straight line AB shall be an asymptote, and the asymptotic area BAED shall be 
infinite. (f) 

(vi) The arcs lying between any two intersections may vary to any extent, may recede to any 
distances whatever from the axis C AC, and approximate to any arcs of any curves to any degree 
of closeness, cutting them, or touching them, or osculating them, at any points and in any manner. 

119. Now, as regards the multiplicity of forces which at the present time physicists call Resolution of the 
by different names, it can also here be observed that, if anyone wants to consider one of these U j^f of N^wtonUn 
separately, the curve though it is of itself quite one-fold can yet be divided into several attraction of 
parts by a sort of mental & fictitious resolution of the forces. Thus, for instance, if f^fother 1 force* 1 d 
anyone wishes to consider universal gravitation of matter exactly reciprocal to the squares 
of the distances ; he can indeed describe on the attractive side the hyperbola which has 
its ordinates accurately in the inverse ratio of the squares of the distances, & this will be 
as it were a continuation of the branch VTS. Then he can add on to every ordinate, such 
as ag, dh, the ordinates of this new hyperbola, in the direction of AB, starting in each case 
from points on the curve, as g,h ; & in this way there will be obtained a fresh curve, which 
for the part pV will approximately coincide with the axis 0C, & for the remainder will 
recede from it & wind itself about it, if the vertices F,K,O are more distant from the 
axis than the corresponding point on the hyperbola. Then it can be stated that all points 
of matter have gravitation accurately decreasing in the inverse square of the distance, 
together with another force represented by this new curve. For it comes to the same 
thing to think of these two laws of forces acting together as of the single law already 
given ; & the results that arise will be the same also. 

1 20. In the same manner this new curve can be divided into two others, or several The resolution of 
others, by considering some other force, in some way or other accurately obeying certain ^o several other 
fixed laws, & at the same time altering the curve just obtained by translating the points of it forces. 
through intervals equal to the ordinates corresponding to the new law that has been taken. 
In this manner several different forces will be obtained ; & this will be sometimes useful, 
as we mentioned that it would be in resolution of forces, for determining their effects more 
readily ; & will be a sort of true resolution of forces. But all this will be as it were only 
a conception of our mind ; for, in reality, there is a single law of forces, & that is the one 
which I gave in Fig. i, & it will be the compounded resultant of all such forces as the above. 

121. Moreover, since I here make mention of gravitation decreasing accurately in the The .. t. heor .y * 

, r11 . .. 111 1111 gravitation is not 

inverse ratio ot the squares ot the distances, it is to be remarked that no one should make in opposition ; this 

any difficulty over the fact that, amongst physicists & more especially those who deal with l! J^ 'H d t es v not hol n 

celestial mechanics, it is considered as an established fact that gravitation decreases accurately distances. 

in the inverse ratio of the squares of the distances, whilst in my Theory the law of forces 

is very different from this ratio. Especially, in the case of extremely small distances, the 

whole force, which the particles exert upon one another, will differ very much in every 

case from the force of gravity, if that is supposed to be inversely proportional to the 

squares of these distances. For, in the case of gases, which exercise such a mighty 

force of self-expansion, there is certainly repulsion at those very small distances from one 

another, & not attraction ; again, the attraction that arises in cohesion is immensely 

greater than it ought to be according to the law of universal gravitation. Now, from the 

results obtained by Newton, the attraction corresponding to gravitation in homogeneous 

spheres of different diameters varies as the diameters of the spheres ; & therefore this 

kind of force for the case of a tiny particle is as small in proportion to the gravitation of 

bodies to the Earth as the diameter of the particle is small in proportion to the diameter 

of the whole Earth ; & is thus insensible altogether. Hence Newton admitted another 

force in the case of cohesion, decreasing in a greater ratio than the inverse square of the 

distances ; also many of the followers of Newton have admitted a force corresponding to 

the formula, a'x 3 + b'x 2 ; in this the first term is immensely less than the second, when x 

is immensely greater than some distance assumed as unit distance ; & immensely greater, 

when x is immensely less. By this means, at sufficiently great distances the first part 

practically vanishes & the force remains very approximately in the inverse ratio of the squares 

of the distances x ; whilst, at very small distances, it is very nearly in the inverse ratio 

of the cubes of the distances. Thus indeed, not even amongst the followers of Newton has 

the inverse ratio of the squares of the distances been altogether rigidly adhered to. 

122. Now Newton proved, in the case of planetary elliptic orbits, that that which The law follows 
Astronomers call the apsidal line, i.e., the axis of the ellipse, would have a very great motion, not 7 


if the ratio of the forces varied to any great extent from the inverse ratio of the squares from the apheiia of 
of the distances ; & since as far as could be observed the lines of apses were stationary 

(f) This is required because in Mechanics it is shown that the area of a curve, whose abscissa r'present distances 
y ordinates forces, represents the increase or decrease of the square of the velocity. Hence in order that the forces 
should be capable of destroying any velocity however great, this area must be greater than any finite area. 


in earum orbitis apsidum linese, intulit, earn rationem observari omnino in gravitate. At 
id nequaquam evincit, accurate servari illam legem, sed solum proxime, neque inde ullum 
efficax argumentum contra meam Theoriam deduci potest. Nam inprimis nee omnino 
quiescunt illae apsidum lineae, sive, quod idem est, aphelia planetarum, sed motu exiguo 
quidem, at non insensibili prorsus, moventur etiam respectu fixarum, adeoque motu non 
tantummodo apparente, sed vero. Tribuitur is motus perturbationi virium ortae ex mutua 
planetarum actione in se invicem ; at illud utique hue usque nondum demonstratum est, 
ilium motum accurate respondere actionibus reliquorum planetarum agentium in ratione 
reciproca duplicata distantiarum ; neque enim adhuc sine contemptibus pluribus, & 
approximationibus a perfectione, & exactitudine admodum remotis solutum est problema, 
quod appellant, trium corporum, quo quasratur motus trium corporum in se mutuo 
agentium in ratione reciproca duplicata distantiarum, & utcunque projectorum, ac illae 
ipsae adhuc admodum imperfectae solutiones, quae prolatae hue usque sunt, inserviunt 
tantummodo particularibus quibusdam casibus, ut ubi unum corpus sit maximum, & 
remotissimum, quemadmodum Sol, reliqua duo admodum minora & inter se proxima, ut 
est Luna, ac Terra, vel remota admodum a majore, & inter se, ut est Jupiter, & Saturnus. 
Hinc nemo hucusque accuratum instituit, aut etiam instituere potuit calculum pro actione 
perturbativa omnium planetarum, quibus si accedat actio perturbativa cometarum, qui, 
nee scitur, quam multi sint, nee quam longe abeant ; multo adhuc magis evidenter patebit, 
nullum inde confici posse argumentum [56] pro ipsa penitus accurata ratione reciproca 
duplicata distantiarum. 

I2 3- Clairautius quidem in schediasmate ante aliquot annos impresso, crediderat, ex 
autem hanc legem ipsis motibus Kneje apsidum Lunae colligi sensibilem recessum a ratione reciproca duplicata 
a uantum iftmerit[ m distantiae, & Eulerus in dissertatione De Aberrationibus Jovis, W Saturni, quas premium 
retulit ab Academia Parisiensi an. 1748, censuit, in ipso Jove, & Saturno haberi recessum 
admodum sensibilem ab ilia ratione ; sed id quidem ex calculi defectu non satis product! 
sibi accidisse Clairautius ipse agnovit, ac edidit ; & Eulero aliquid simile fortasse accidit : 
nee ullum habetur positivum argumentum pro ingenti recessu gravitatis generalis a ratione 
duplicata distantiarum in distantia Lunae, & multo magis in distantia planetarum. Vero 
nee ullum habetur argumentum positivum pro ratione ita penitus accurata, ut discrimen 
sensum omnem prorsus effugiat. At & si id haberetur ; nihil tamen pati posset inde 
Theoria mea ; cum arcus ille meae curvae postremus VT possit accedere, quantum libuerit, 
ad arcum illius hyperbolae, quae exhibet legem gravitatis reciprocam quadratorum dis- 
tantiae, ipsam tangendo, vel osculando in punctis quotcunque, & quibuscunque ; adeoque 
ita possit accedere, ut discrimen in iis majoribus distantiis sensum omnem effugiat, & 
effectus nullum habeat sensibile discrimen ab effectu, qui responderet ipsi legi gravitatis ; 
si ea accurate servaret proportionem cum quadratis distantiarum reciproce sumptis. 

Difficuitas a Mau- 124. Nee vero quidquam ipsi meae virium Theorias obsunt meditationes Maupertuisii, 

tionemaxfma^Nlw- ingeniosae illae quidem, sed meo judicio nequaquam satis conformes Natune legibus circa 

tonianae legis. legem virium decrescentium in ratione reciproca duplicata distantiarum, cujus ille perfec- 

tiones quasdam persequitur, ut illam, quod in hac una integri globi habeant eandem virium 

legem, quam singulae particulae. Demonstravit enim Newtonus, globos, quorum singuli 

paribus a centre distantiis homogenei sint, & quorum particulae minimae se attrahant in 

ratione reciproca duplicata distantiarum, se itidem attrahere in eadem ratione distantiarum 

reciproca duplicata. Ob hasce perfectiones hujus Theoriae virium ipse censuit hanc legem 

reciprocam duplicatam distantiarum ab Auctore Naturae selectam fuisse, quam in Natura 

esse vellet. 

Prima responsio : 125. At mihi quidem inprimis nee unquam placuit, nee placebit sane unquam in 

n !^Jf 8 T s ^rwt 8 investieatione Naturae causarum fmalium usus, quas tantummodo ad meditationem quandam, 

onmcs, *x jjcricui~ o f i i t XT 1*1* * "VT 

iones, ac seiigi et- contemplationemque, usui esse posse abitror, ubi leges JNaturse aliunde innotuennt. JNam 
^J$L*5* nee perfectiones omnes innotescere nobis possunt, qui intimas rerum naturas nequaquam 

III grH.ilclIU pcrlcC~ * f . * a C 

tionum. inspicimus, sed externas tantummodo propnetates quasdam agnoscimus, & lines omnes, 

quos Naturae Auctor sibi potuit [57] proponere, ac proposuit, dum Mundum conderet, 


in the orbits of each, he deduced that the ratio of the inverse square of the distances was 
exactly followed in the case of gravitation. But he only really proved that that law was 
very approximately followed, & not that it was accurately so ; nor from this can any 
valid argument against my Theory be brought forward. For, in the first place these lines 
of apses, or what comes to the same thing, the aphelia of the planets are not quite stationary ; 
but they have some motion, slight indeed but not quite insensible, with respect to the fixed 
stars, & therefore move not only apparently but really. This motion is attributed to 
the perturbation of forces which arises from the mutual action of the planets upon one 
another. But the fact remains that it has never up till now been proved that this motion 
exactly corresponds with the actions of the rest of the planets, where this is in accordance 
with the inverse ratio of the squares of the distances. For as yet the problem of three bodies, 
as they call it, has not been solved except by much omission of small quantities & by 
adopting approximations that are very far from truth and accuracy ; in this problem is 
investigated the motion of three bodies acting mutually upon one another in the inverse 
ratio of the squares of the distances, & projected in any manner. Moreover, even these 
still only imperfect solutions, such as up till now have been published, hold good only 
in certain particular cases ; such as the case in which one of the bodies is very large & at 
a very great distance, the Sun for instance, whilst the other two are quite small in comparison 
& very near one another, as are the Earth and the Moon, or at a large distance from the 
greater & from one another as well, as Jupiter & Saturn. Hence nobody has hitherto 
made, nor indeed could anybody make, an accurate calculation of the disturbing influence 
of all the other planets combined. If to this is added the disturbing influence of the comets, 
of which we neither know the number, nor how far off they are ; it will be still more evident 
that from this no argument can be built up in favour of a perfectly exact observance of 
the inverse ratio of the squares of the distances. 

123. Clairaut indeed, in a pamphlet printed several years ago, asserted his belief that The same thing is 
he had obtained from the motions of the line of apses for the Moon a sensible discrepancy J? ^ ^duced from 

, , . r i i AIT--I i i r>^r T the res t of astro- 

from the inverse square of the distance. Also Euler, in his dissertation De Aberratiombus nomy ; moreover 
Jovis, y Saturni, which carried off the prize given by the Paris Academy, considered that thls Iaw of . mi 1 e 

/, ,. T . <-, , r . ' .. , ,. *', can approximate 

in the case of Jupiter & Saturn there was quite a sensible discrepancy from that ratio, to the other as 

But Clairaut found out, & proclaimed the fact, that his result was indeed due to a defect nearl y as is desired. 

in his calculation which had not been carried far enough ; & perhaps something similar 

happened in Euler's case. Moreover, there is no positive argument in favour of a large 

discrepancy from the inverse ratio of the squares of the distances for universal gravitation 

in the case of the distance of the Moon, & still more in the case of the distances of the planets. 

Neither is there any rigorous argument in favour of the ratio being so accurately observed 

that the difference altogether eludes all observation. But even if this were the case, my 

Theory would not suffer in the least because of it. For the last arc VT of my curve can 

be made to approximate as nearly as is desired to the arc of the hyperbola that represents 

the law of gravitation according to the inverse squares of the distances, touching the latter, 

or osculating it in any number of points in any positions whatever ; & thus the approximation 

can be made so close that at these relatively great distances the difference will be altogether 

unnoticeable, & the effect will not be sensibly different from the effect that would 

correspond to the law of gravitation, even if that exactly conformed to the inverse ratio 

of the squares of the distances. 

124. Further, there is nothing really to be objected to my Theory on account of the Objection arising 
meditations of Maupertuis ; these are certainly most ingenious, but in my opinion in no p r ct ion fccord* 
way sufficiently in agreement with the laws of Nature. Those meditations of his, I mean, ing to Maupertuis, 
with regard to the law of forces decreasing in the inverse ratio of the squares of the distances ; j^ f w the Newtoman 
for which law he strives to adduce certain perfections as this, that in this one law alone 

complete spheres have the same law of forces as the separate particles of which they are 
formed. For Newton proved that spheres, each of which have equal densities at equal 
distances from the centre, & of which the smallest particles attract one another in the 
inverse ratio of the squares of the distances, themselves also attract one another in the same 
ratio of the inverse squares of the distances. On account of such perfections as these in 
this Theory of forces, Maupertuis thought that this law of the inverse squares of the distances 
had been selected by the Author of Nature as the one He willed should exist in Nature. 

125. Now, in the first place I was never satisfied, nor really shall I ever be satisfied, First reply to this ; 
with the use of final causes in the investigation of Nature ; these I think can only be employed perfections^ 'not 
for a kind of study & contemplation, in such cases as those in which the laws of Nature known ; and even 
have already been ascertained from other methods. For we cannot possibly be acquainted ^sdlcted^fo^'fhe 
with all perfections ; for in no wise do we observe the inmost nature of things, but all we sake of greater per- 
know are certain external properties. Nor is it at all possible for us to see & know all fl 

the intentions which the Author of Nature could and did set before Himself when He founded 


videre, & nosse omnino non possumus. Quin immo cum juxta ipsos Leibnitianos inprimis, 
aliosque omnes defensores acerrimos principii rationis sufficients, & Mundi perfectissimi, 
qui inde consequitur, multa quidem in ipso Mundo sint mala, sed Mundus ipse idcirco 
sit optimus, quod ratio boni ad malum in hoc, qui electus est, omnium est maxima ; fieri 
utique poterit, ut in ea ipsius Mundi parte, quam hie, & nunc contemplamur, id, quod 
electum fuit, debuerit esse non illud bonum, in cujus gratiam tolerantur alia mala, sed 
illud malum, quod in aliorum bonorum gratiam toleratur. Quamobrem si ratio reciproca 
duplicata distantiarum esset omnium perfectissima pro viribus mutuis particularum, non 
inde utique sequeretur, earn pro Natura fuisse electam, & constitutam. 

Eandem legem nee I2 6. At nee revera perfectissima est, quin immo meo quidem judicio est omnino 

pcrfcctam esse, nee r v 1 i ... 

in corporibus, non imperfecta, & tarn ipsa, quam aliae plunmse leges, quas requirunt attractionem immmutis 
utique accurate distantiis crcscentcm in ratione reciproca duplicata distantiarum, ad absurda deducunt 
' plurima, vel saltern ad inextricabiles difficultates, quod ego quidem turn alibi etiam, turn 
inprimis demonstravi in dissertatione De Lege Firium in Natura existentium a num. 59. (g) 
Accedit autem illud, quod ilia, qua; videtur ipsi esse perfectio maxima, quod nimirum 
eandem sequantur legem globi integri, quam particulae minimae, nulli fere usui est in 
Natura ; si res accurate ad exactitudinem absolutam exigatur ; cum nulli in Natura sint 
accurate perfecti globi paribus a centre distantiis homogenei, nam praeter non exiguam 
inaequalitatem interioris textus, & irregularitatem, quam ego quidem in Tellure nostra 
demonstravi in Opere, quod de Litteraria Expeditione per Pontificiam ditionem inscripsi, 
in reliquis autem planetis, & cometis suspicari possumus ex ipsa saltern analogia, prater 
scabritiem superficiei, quaj utique est aliqua, satis patet, ipsa rotatione circa proprium 
axem induci in omnibus compressionem aliquam, quae ut ut exigua, exactam globositatem 
impedit, adeoque illam assumptam perfectionem maximam corrumpit. Accedit autem 
& illud, quod Newtoniana determinatio rationis reciprocal duplicatae distantiarum locum 
habet tantummodo in globis materia continua constantibus sine ullis vacuolis, qui globi 
in Natura non existunt, & multo minus a me admitti possunt, qui non vacuum tantummodo 
disseminatum in materia, ut Philosophi jam sane passim, sed materiam in immenso vacuo 
innatantem, & punctula a se invicem remota, ex quibus, qui apparentes globi fiant, illam 
habere proprietatem non possunt rationis reciprocal duplicatae distantiarum, adeoque nee 
illius perfectionis creditas maxime perfectam, absolutamque applicationem. 

o ex prae- \<:$\ \2j. Demum & illud nonnullis difficultatem parit summam in hac Theoria 

juuiv-.w pro impul- ~ * ' . . . . . . f i. i i .. 

sione, & ex testi- Virium, quod censeant, phaenomena omnia per impulsionem explicari debere, & immedi- 
monio sensuum : a tum contactum, quern ipsum credant evidenti sensuum testimonio evinci ; hinc huiusmodi 

responsio ad hanc . r n XT i 

posteriorem. nostras vires immechamcas appellant, & eas, ut & Newtomanorum generalem gravitatem, 

vel idcirco rejiciunt, quod mechanicae non sint, & mechanismum, quem Newtoniana 
labefactare coeperat, penitus evertant. Addunt autem etiam per jocum ex serio argumento 
petito a sensibus, baculo utendum esse ad persuadendum neganti contactum. Quod ad 
sensuum testimonium pertinet, exponam uberius infra, ubi de extensione agam, quae eo 
in genere habeamus praejudicia, & unde : cum nimirum ipsis sensibus tribuamus id, 
quod nostrae ratiocinationis, atque illationis vitio est tribuendum. Satis erit hie monere 
illud, ubi corpus ad nostra organa satis accedat, vim repulsivam, saltern illam ultimam, 
debere in organorum ipsorum fibris excitare motus illos ipsos, qui excitantur in 
communi sententia ab impenetrabilitate, & contactu, adeoque eundem tremorem ad 
cerebrum propagari, & eandem excitari debere in anima perceptionem, quae in 
communi sententia excitaretur ; quam ob rem ab iis sensationibus, quae in hac ipsa 
Theoria Virium haberentur, nullum utique argumentum desumi potest contra ipsam, 
quod ullam vim habeant utcunque tenuem. 

Felicius explicari 128. Quod pertinet ad explicationem phaenomenorum per impulsionem immediatam, 

sione*- "eam^nus- rnonui sane superius, quanto felicius, ea prorsus omissa, Newtonus explicarit Astronomiam, 

quam positive pro- & Opticam ; & patebit inferius, quanto felicius phaenomena quaeque praecipua sine ulla 

immediata impulsione explicentur. Cum iis exemplis, turn aliis, commendatur abunde 

ea ratio explicandi phsenomena, quae adhibet vires agentes in aliqua distantia. Ostendant 

(g) Qute hue pertinent, (J continentur novem numeris ejus Dissertations incipiendo a 59, habentur in fine Supplem. 


the Universe. Nay indeed, since in the doctrine of the followers of Leibniz more especially, 
and of all the rest of the keenest defenders of the principle of sufficient reason, and a most 
perfect Universe which is a direct consequence of that idea, there may be many evils in the 
Universe, and yet the Universe may be the best possible, just because the ratio of 
good to evil, in this that has been chosen, is the greatest possible. It might certainly happen 
that in this part of the Universe, which here & now we are considering, that which was 
chosen would necessarily be not that goodness in virtue of which other things that are 
evil are tolerated, but that evil which is tolerated because of the other things that are good. 
Hence, even if the inverse ratio of the squares of the distances were the most perfect of all 
for the mutual forces between particles, it certainly would not follow from that fact that 
it was chosen and established for Nature. 

126. But this law as a matter of fact is not the most perfect of all; nay rather, in This law is neither 
my opinion, it is altogether imperfect. Both it, & several other laws, that require ^ ec ^Tor i0 Dod t - 
attraction at very small distances increasing in the inverse ratio of the squares of the distances ies that are not 
lead to very many absurdities ; or at least, to insuperable difficulties, as I showed in the exactl y spherical, 
dissertation De Lege Virium in Natura existentium in particular, as well as in other places. (g) 

In addition there is the point that the thing, which to him seems to be the greatest 
perfection, namely, the fact that complete spheres obey the same law as the smallest 
particles composing them, is of no use at all in Nature ; for there are in Nature no exactly 
perfect spheres having equal densities at equal distances from the centre. Besides the 
not insignificant inequality & irregularity of internal composition, of which I proved the 
existence in the Earth, in a work which I wrote under the title of De Litteraria Ex-peditione 
per Pontificiam ditionem, we can assume also in the remaining planets & the comets (at 
least by analogy), in addition to roughness of surface (of which it is sufficiently evident that 
at any rate there is some), that there is some compression induced in all of them by the 
rotation about their axes. This compression, although it is indeed but slight, prevents 
true sphericity, & therefore nullifies that idea of the greatest perfection. There is too 
the further point that the Newtonian determination of the inverse ratio of the squares 
of the distances holds good only in spheres made up of continuous matter that is free from 
small empty spaces ; & such spheres do not exist in Nature. Much less can I admit 
such spheres ; for I do not so much as admit a vacuum disseminated throughout matter, 
as philosophers of all lands do at the present time, but I consider that matter as it were 
swims in an immense vacuum, & consists of little points separated from one another. 
These apparent spheres, being composed of these points, cannot have the property of the 
inverse ratio of the squares of the distances ; & thus also they cannot bear the true & 
absolute application of that perfection that is credited so highly. 

127. Finally, some persons raise the greatest objections to this Theory of mine, because Objection founded 
they consider that all the phenomena must be explained by impulse and immediate contact ; "mpui^and'on the 
this they believe to be proved by the clear testimony of the senses. So they call forces testimony of the 
like those I propose non-mechanical, and reject them, just as they also reject the universal th?s latter. rep ' y t0 
gravitation of Newton, for the alleged reason that they are not mechanical, and overthrow 

altogether the idea of mechanism which the Newtonian theory had already begun to 
undermine. Moreover, they also add, by way of a joke in the midst of a serious argument 
derived from the senses, that a stick would be useful for persuading anyone who denies 
contact. Now as far as the evidence of the senses is concerned, I will set forth below, 
when I discuss extension, the prejudices that we may form in. such cases, and the origin 
of these prejudices. Thus, for instance, we may attribute to the senses what really ought 
to be attributed to the imperfection of our reasoning and inference. It will be enough 
just for the present to mention that, when a body approaches close enough to our organs, 
my repulsive force (at any rate it is that finally), is bound to excite in the nerves of those 
organs the motions which, according to the usual idea, are excited by impenetrability and 
contact ; & that thus the same vibrations are sent to the brain, and these are bound to 
excite the same perception in the mind as would be excited in accordance with the usual 
idea. Hence, from these sensations, which are also obtained in my Theory of Forces, no 
argument can be adduced against the theory, which will have even the slightest validity. 

128. As regards the explanation of phenomena by means of immediate contact I, h s a ver ^ thin e g is ^^ 
indeed, mentioned above how much more happily Newton had explained Astronomy and without the idea of 
Optics by omitting it altogether ; and it will be evident, in what follows, how much more |^^ lse ^ nowhere 
happily every one of the important phenomena is explained without any idea of immediate rigorously proved 
contact. - Both by these instances, and by many others, this method of explaining phenomena, to exist - 

by employing forces acting at a distance, is strongly recommended. Let objectors bring 

(g) That which refers to this point, & which is contained in nine articles of the dissertation commencing with Art. 59, 
is to bf found at the end of this work as Supplement IV, 


isti vel unicum exemplum, in quo positive probare possint, per immediatam impulsionem 
communicari motum in Natura. Id sane ii praestabunt nunquam ; cum oculorum testi- 
monium ad excludendas distantias illas minimas, ad quas primum crus repulsivum pertinet, 
& contorsiones curvae circa axem, quae oculos necessario fugiunt, adhibere non possint ; cum 
e contrario ego positive argumento superius excluserim immediatum contactum omnem, 
& positive probaverim, ipsum, quern ii ubique volunt, haberi nusquam. 

Vires hujus Theo- I2 g j) e no minibus quidem non esset, cur solicitudinem haberem ullam ; sed ut & 

rise pertineread ve- ...,*,.., . ~t . . , ' . . ,. ... 

rum, nee occuitum in nsdem aliquid prasjudicio cmdam, quod ex communi loquendi usu provenit, mud 
mechanismum. notandum duco, Mechanicam non utique ad solam impulsionem immediatam fuisse 
restrictam unquam ab iis, qui de ipsa tractarunt, sed ad liberos inprimis adhibitam contem- 
plandos motus, qui independenter ab omni impulsione habeantur. Quae Archimedes de 
aequilibrio tradidit, quse Galilaeus de li-[59]-bero gravium descensu, ac de projectis, quae 
de centralibus in circulo viribus, & oscillationis centre Hugenius, quae Newtonus generaliter 
de motibus in trajectoriis quibuscunque, utique ad Mechanicam pertinent, & Wolfiana 
& Euleriana, & aliorum Scriptorum Mechanica passim utique ejusmodi vires, & motus inde 
ortos contemplatur, qui fiant impulsione vel exclusa penitus, vel saltern mente seclusa. 
Ubicunque vires agant, quae motum materiae gignant, vel immutent, & leges expandantur, 
secundum quas velocitas oriatur, mutetur motus, ac motus ipse determinetur ; id omne 
inprimis ad Mechanicam pertinet in admodum propria significatione acceptam. Quam- 
obrem ii maxime ea ipsa propria vocum significatione abutuntur, qui impulsionem unicam 
ad Mechanismum pertinere arbitrantur, ad quern haec virium genera pertinent multo magis, 
qu33 idcirco appellari jure possunt vires Mechanic*?, & quidquid per illas fit, jure affirmari 
potest fieri per Mechanismum, nee vero incognitum, & occuitum, sed uti supra demonstra- 
vimus, admodum patentem, a manifestum. 

Discrimen inter j -m E o dem etiam pacto in omnino propria significatione usurpare licebit vocem con- 

contactum mathe- J . . .. * T i i 

maticum, & physi- tactus ; licet intervallum semper remaneat aliquod ; quanquam ego ad aequivocationes evi- 
cum : hunc did tandas soleo distinguere inter contactum Mathematicum, in quo distantia sit prorsus nulla, 

proprie contactum. ni j- a. o 1 

& contactum Physicum, in quo distantia sensus effugit omnes, & vis repulsiva satis magna 
ulteriorem accessum per nostras vires inducendum impedit. Voces ab hominibus institutae 
sunt ad significandas res corporeas, & corporum proprietates, prout nostris sensibus subsunt, 
iis, quae continentur infra ipsos, nihil omnino curatis. Sic planum, sic laeve proprie dicitur 
id, in quo nihil, quod sensu percipi possit, sinuetur, nihil promineat ; quanquam in communi 
etiam sententia nihil sit in Natura mathematice planum, vel laeve. Eodem pacto & nomen 
contactus ab hominibus institutum est, ad exprimendum physicum ilium contactum tantum- 
modo, sine ulla cura contactus mathematics, de quo nostri sensus sententiam ferre non 
possunt. Atque hoc quidem pacto si adhibeantur voces in propria significatione ilia, quae 
ipsarum institutioni respondeat ; ne a vocibus quidem ipsis huic Theoriae virium invidiam 
creare poterunt ii, quibus ipsa non placet. 

extensionis sit orta. 

Transitus ab ob- j^j. Atque haec de iis, quae contra ipsam virium legem a me propositam vel objecta 

Theoriam virium sunt hactenus, vel objici possent, sint satis, ne res in infinitum excrescat. Nunc ad ilia 
ad objections con- transibimus, quae contra constitutionem elementorum materiae inde deductam se menti 

tra puncta. .*.... i . j. 

oiferunt, in quibus itidem, quae maxime notatu digna sunt, persequar. 

Objectio ab idea 132. Inprimis quod pertinet ad hanc constitutionem elementorum materise, sunt 

puncti inextensi, multi, qui nullo pacto in animum sibi possint inducere, ut admittant puncta prorsus 

qua caremus : re- , r i n T 11 j A - J 

sponsio : unde idea mdi-[6o]-visibiha, & mextensa, quod nullam se dicant habere posse eorum ideam. At id 
a - hominum genus praejudiciis quibusdam tribuit multo plus aequo. Ideas omnes, saltern 
eas, quae ad materiam pertinent, per sensus hausimus. Porro sensus nostri nunquam 
potuerunt percipere singula elementa, quae nimirum vires exerunt nimis tenues ad movendas 
fibras, & propagandum motum ad cerebrum : massis indiguerunt, sive elementorum 
aggregatis, quae ipsas impellerent collata vi. Haec omnia aggregata constabant partibus, 
quarum partium extremae sumptae hinc, & inde, debebant a se invicem distare per aliquod 
intervallum, nee ita exiguum. Hinc factum est, ut nullam unquam per sensus acquirere 
potuerimus ideam pertinentem ad materiam, quae simul & extensionem, & partes, ac 
divisibilitatem non involverit. Atque idcirco quotiescunque punctum nobis animo sistimus, 
nisi reflexione utamur, habemus ideam globuli cujusdam perquam exigui, sed tamen globuli 
rotundi, habentis binas superficies oppositas distinctis. 


forward but a single instance in which they can positively prove that motion in Nature 
is communicated by immediate impulse. Of a truth they will never produce one ; for 
they cannot use the testimony of the eyes to exclude those very small distances to which 
the first repulsive branch of my curve refers & the windings about the axis ; for these 
necessarily evade ocular observation. Whilst I, on the other hand, by the rigorous argument 
given above, have excluded all idea of immediate contact ; & I have positively proved 
that the thing, which they wish to exist everywhere, as a matter of fact exists nowhere. 

129. There is no reason why I should trouble myself about nomenclature ; but, as The forces in this 
in that too there is something that, from the customary manner of speaking, gives rise to ^j^/^ot to an 
a kind of prejudice, I think it should be observed that Mechanics was certainly never occult mechanism, 
restricted to immediate impulse alone by those who have dealt with it ; but that in the 

first place it was employed for the consideration of free motions, such as exist quite 
independently of any impulse. The work of Archimedes on equilibrium, that of Galileo 
on the free descent of heavy bodies & on projectiles, that of Huygens on central forces 
in a circular orbit & on the centre of oscillation, what Newton proved in general for 
motion on all sorts of trajectories ; all these certainly belong to the science of Mechanics. 
The Mechanics of Wolf, Euler & other writers in different lands certainly treats of such 
forces as these & the motions that arise from them, & these matters have been accomplished 
with the idea of impulse excluded altogether, or at least put out of mind. Whenever 
forces act, & there is an investigation of the laws in accordance with which velocity is 
produced, motion is changed, or the motion itself is determined ; the whole of this belongs 
especially to Mechanics in a truly proper signification of the term. Hence, they greatly 
abuse the proper signification of terms, who think that impulse alone belongs to the science 
of Mechanics ; to which these kinds of forces belong to a far greater extent. Therefore 
these forces may justly be called Mechanical ; & whatever comes about through their 
action can be justly asserted to have come about through a mechanism ; & one too that 
is not unknown or mysterious, but, as we proved above, perfectly plain & evident. 

130. Also in the same way we may employ the term contact in an altogether special Distinction be- 
sense ; the interval may always remain something definite. Although, in order to avoid ticai n and m ph^ e skai 
ambiguity, I usually distinguish between mathematical contact, in which the distance is contact ; the latter 
absolutely nothing, & -physical contact, in which the distance is too small to affect our C anedorftact > . per y 
senses, and the repulsive force is great enough to prevent closer approach being induced 

by the forces we are considering. Words are formed by men to signify corporeal things 
& the properties of such, as far as they come within the scope of the senses ; & those 
that fall beneath this scope are absolutely not heeded at all. Thus, we properly call a 
thing plane or smooth, which has no bend or projection in it that can be perceived by the 
senses ; although, in the general opinion, there is nothing in Nature that is mathematically 
plane or smooth. In the same way also, the term contact was invented by men to express 
physical contact only, without any thought of mathematical contact, of which our senses 
can form no idea. In this way, indeed, if words are used in their correct sense, namely, 
that which corresponds to their original formation, those who do not care for my Theory 
of forces cannot from those words derive any objection against it. 

131. I have now said sufficient about those objections that either up till now have Passing on from 
been raised, or might be raised, against the law of forces that I have proposed ; otherwise ^y^Theorf ""of 
the matter would grow beyond all bounds. Now we will pass on to objections against forces to objections 
the constitution of the elements of matter derived from it, which present themselves to the a g amst P mts - 
mind ; & in these also I will investigate those that more especially seem worthy of remark. 

132. First of all, as regards the constitution of the elements of matter, there are indeed Potion to^the 
many persons who cannot in any way bring themselves into that frame of mind to admit tended points, 
the existence of points that are perfectly indivisible and non-extended ; for they say that which we postu- 

* < ' T* i r 1_ 13. tc , reply , tiic 

they cannot form any idea of such points. But that type of men pays more heed than origin of the idea 

is right to certain prejudices. We derive all our ideas, at any rate those that relate to of extension. 

matter, from the evidences of our senses. Further, our senses never could perceive single 

elements, which indeed give forth forces that are too slight to affect the nerves & thus 

propagate motion to the brain. The senses would need masses, or aggregates of the elements, 

which would affect them as a result of their combined force. Now all these aggregates are 

made up of parts ; & of these parts the two extremes on the one side and on the^ other 

must be separated from one another by a certain interval, & that not an insignificant 

one. Hence it comes about that we could never obtain through the senses any idea relating 

to matter, which did not involve at the same time extension, parts & divisibility. So, 

as often as we thought of a point, unless we used our reflective powers, we should get the 

idea of a sort of ball, exceedingly small indeed, but still a round ball, having two distinct 

and opposite faces. 


idea m puncti 133. Quamobrem ad concipiendum punctum indivisibile, & inextensum ; non debemus 

refl^xionemT'quo- consulere ideas > q uas immediate per sensus hausimus ; sed earn nobis debemus efformare 

modo ejus idea per reflexionem. Reflexione adhibita non ita difficulter efformabimus nobis ideam ejusmodi. 

negativa acqmra- N am i n p r imi s u bi & extensionem, & partium compositionem conceperimus ; si utranque 

negemus ; jam inextensi, & indivisibilis ideam quandam nobis comparabimus per negati- 

onem illam ipsam eorum, quorum habemus ideam ; uti foraminis ideam habemus utique 

negando existentiam illius materias, quas deest in loco foraminis. 

Quomodo ejus idea 134. Verum & positivam quandam indivisibilis, & inextensi puncti ideam poterimus 

posfit^per itmlte" com P arare n bis ope Geometrias, & ope illius ipsius ideas extensi continui, quam per sensus 
& limitum inter- hausimus, & quam inferius ostendemus, fallacem esse, ac fontem ipsum fallacies ejusmodi 
aperiemus, quas tamen ipsa ad indivisibilium, & inextensorum ideam nos ducet admodum 
claram. Concipiamus planum quoddam prorsus continuum, ut mensam, longum ex. gr. 
pedes duos ; atque id ipsum planum concipiamus secari transversum secundum longitudinem 
ita, ut tamen iterum post sectionem conjungantur partes, & se contingant. Sectio ilia 
erit utique limes inter partem dexteram & sinistram, longus quidem pedes duos, quanta 
erat plani longitude, at latitudinis omnino expers : nam ab altera parte immediate motu 
continue transitur ad alteram, quse, si ilia sectio crassitudinem haberet aliquam, non esset 
priori contigua. Ilia sectio est limes secundum crassitudinem inextensus, & indivisibilis, 
cui si occurrat altera sectio transversa eodem pacto indivisibilis, & inextensa ; oportebit 
utique, intersectio utriusque in superficie plani concepti nullam omnino habeat extensionem 
in partem quamcumque. Id erit punctum peni-[6i]-tus indivisibile, & inextensum, quod 
quidem punctum, translate piano, movebitur, & motu suo lineam describet, longam quidem, 
sed latitudinis expertem. 

Natura inextensi, j^c. Quo autem melius ipsius indivisibilis natura concipi possit ; quasrat a nobis 

quod non potest . /" r , . , . ^. . . ' ". 

esse inextenso con- quispiam, ut aliam faciamus ejus planae massas sectionem, quas priori ita sit proxima, ut 
tiguum in Uneis. n ihil prorsus inter utramque intersit. Respondebimus sane, id fieri non posse : vel enim 
inter novam sectionem, & veteram intercedet aliquid ejus materias, ex qua planum con- 
tinuum constare concipimus, vel nova sectio congruet penitus cum praecedente. En 
quomodo ideam acquiremus etiam ejus naturas indivisibilis illius, & inextensi, ut aliud 
indivisibile, & inextensum ipsi proximum sine medio intervallo non admittat, sed vel cum 
eo congruat, vel aliquod intervallum relinquat inter se, & ipsum. Atque hinc patebit 
etiam illud, non posse promoveri planum ipsum ita, ut ilia sectio promoveatur tantummodo 
per spatium latitudinis sibi asqualis. Utcunque exiguus fuerit motus, jam ille novus 
sectionis locus distabit a praecedente per aliquod intervallum, cum sectio sectioni contigua 
esse non possit. 

Eademin punctis : 136. Hasc si ad concursum sectionum transferamus, habebimus utique non solum ideam 

idea puncti eeo- ..,...,.,.. . , . ,. ... v j -i 

metricf transiata puncti indivisibilis, & inextensi, sed ejusmodi naturae puncti ipsius, ut aliud punctum sibi 
ad physicum, & contiguum habere non possit, sed vel congruant, vel aliquo a se invicem intervallo distent. 
Et hoc pacto sibi & Geometrae ideam sui puncti indivisibilis, & inextensi, facile efformare 
possunt, quam quidem etiam efformant sibi ita, ut prima Euclidis definitio jam inde incipiat : 
punctum est, cujus nulla pars est. Post hujusmodi ideam acquisitam illud unum intererit 
inter geometricum punctum, & punctum physicum materiae, quod hoc secundum habebit 
proprietates reales vis inertias, & virium illarum activarum, quas cogent duo puncta ad se 
invicem accedere, vel a se invicem recedere, unde net, ut ubi satis accesserint ad organa 
nostrorum sensuum, possint in iis excitare motus, qui propagati ad cerebrum, perceptiones 
ibi eliciant in anima, quo pacto sensibilia erunt, adeoque materialia, & realia, non pure 

Punctorum exist- j-- g n jgjtur per reffexionem acquisitam ideam punctorum realium, materialium, 

entiam aliunde . .. , J .( . . . , . . r r . 

demonstrari : per indivisibilium, inextensorum, quam inter ideas ab infantia acquisitas per sensus mcassum 
ideam acquisitam q uaer i m us. Idea ejusmodi non evincit eorum existentiam. Ipsam quam nobis exhibent 

ea tantum concipi. ^ . . J . , , . . ..*-."... 

positiva argumenta superms facta, quod mmirum, ne admittatur in colhsione corporum 
saltus, quern & inductio, & impossibilitas binarum velocitatum diversarum habendarum 
omnino ipso momento, quo saltus fieret, excludunt, oportet admittere in materia vires, 
quas repulsivae sint in minimis distantiis, & iis in infinitum imminutis augeantur in infinitum ; 


133. Hence for the purpose of forming an idea of a point that is indivisible & non- The idea of a point 
extended, we cannot consider the ideas that we derive directly from the senses ; but we ^"refleTti obtai h rxed 
must form our own idea of it by reflection. If we reflect upon it, we shall form an idea a negative >n ideaof 
of this sort for ourselves without much difficulty. For, in the first place, when we have con- rt may ^ ac( i uired - 
ceived the idea of extension and composition by parts, if we deny the existence of both, then 

we shall get a sort of idea of non-extension & indivisibility by that very negation of the 
existence of those things of which we already have formed an idea. For instance, we have 
the idea of a hole by denying the existence of matter, namely, that which is absent from 
the position in which the hole lies. 

134. But we can also get an idea of a point that is indivisible & non-extended, by HOW a positive idea 
the aid of geometry, and by the help of that idea of an extended continuum that we derive ^^ ^of^bourf 
from the senses ; this we will show below to be a fallacy, & also we will open up the very daries, and inter- 
source of this kind of fallacy, which nevertheless will lead us to a perfectly clear idea of ^g ns of boun " 
indivisible & non-extended points. Imagine some thing that is perfectly plane and 
continuous, like a table-top, two feet in length ; & suppose that this plane is cut across 

along its length ; & let the parts after section be once more joined together, so that they 
touch one another. The section will be the boundary between the left part and the right 
part ; it will be two feet in length (that being the length of the plane before section), & 
altogether devoid of breadth. For we can pass straightaway by a continuous motion 
from one part to the other part, which would not be contiguous to the first part if the section 
had any thickness. The section is a boundary which, as regards breadth, is non-extended 
& indivisible ; if another transverse section which in the same way is also indivisible & 
non-extended fell across the first, then it must come about that the intersection of the 
two in the surface of the assumed plane has no extension at all in any direction. It will 
be a point that is altogether indivisible and non-extended ; & this point, if the plane 
be moved, will also move and by its motion will describe a line, which has length indeed 
but is devoid of breadth. 

135. The nature of an indivisible itself can be better conceived in the following way. The nature of a 
Suppose someone should ask us to make another section of the plane mass, which shall lie hin~ g ext w I hich 
so near to the former section that there is absolutely no distance between them. We cannot he next to 
should indeed reply that it could not be done. For either between the new section & ^ S 

the old there would intervene some part of the matter of which the continuous plane was concerned, 
composed ; or the new section would completely coincide with the first. Now see how 
we acquire an idea also of the nature of that indivisible and non-extended thing, which 
is such that it does not allow another indivisible and non-extended thing to lie next to it 
without some intervening interval ; but either coincides with it or leaves some definite 
interval between itself & the other. Hence also it will be clear that it is not possible 
so to move the plane, that the section will be moved only through a space equal to its own 
breadth. However slight the motion is supposed to be, the new position of the section 
would be at a distance from the former position by some definite interval ; for a section 
cannot be contiguous to another section. 

136. If now we transfer these arguments to the intersection of sections, we shall truly Th . e same thing for 
have not only the idea of an indivisible & non-extended point, but also an idea of the ^geometrical point 
nature of a point of this sort ; which is such that it cannot have another point contiguous transferred to a 
to it, but the two either coincide or else they are separated from one another by some interval. riafpoLt^ 

In this way also geometricians can easily form an idea of their own kind of indivisible & 
non-extended points ; & indeed they do so form their idea of them, for the first defi- 
nition of Euclid begins : A -point is that which has no parts. After an idea of this sort has 
been acquired, there is but one difference between a geometrical point & a physical point 
of matter ; this lies in the fact that the latter possesses the real properties of a force of 
inertia and of the active forces that urge the two points to approach towards, or recede 
from, one another ; whereby it comes about that when they have approached sufficiently 
near to the organs of our senses, they can excite motions in them which, when propagated 
to the brain, induce sensations in the mind, and in this way become sensible, & thus 
material and real, & not imaginary. 

137. See then how by reflection the idea of real, material, indivisible, non-extended The existence of 
points can be acquired ; whilst we seek for it in vain amongst those ideas that we have o^herwise^demon- 
acquired since infancy by means of the senses. But an idea of this sort about things does strated ; they can 
not prove that these things exist. That is just what the rigorous arguments given above through ^cquir- 
point out to us ; that is to say, because, in order that in the collision of solids a sudden ing an idea of them, 
change should not be admitted (which change both induction & the impossibility of 

there being two different velocities at the same instant in which the change should take 
place), it had to be admitted that in matter there were forces which are repulsive ^at very 
small distances, & that these increased indefinitely as the distances were diminished. 



unde fit, ut duse particulae materiae sibi [62] invicem contiguae esse non possint : nam illico 
vi ilia repulsiva resilient a se invicem, ac particula iis constans statim disrumpetur, adeoque 
prima materiae elementa non constant contiguis partibus, sed indivisibilia sunt prorsus, 
atque simplicia, & vero etiam ob inductionem separabilitatis, ac distinctionis eorum, quae 
occupant spatii divisibilis partes diversas, etiam penitus inextensa. Ilia idea acquisita per 
reflexionem illud praestat tantummodo, ut distincte concipiamus id, quod ejusmodi rationes 
ostendunt existere in Natura, & quod sine reflexione, & ope illius supellectilis tantummodo, 
quam per sensus nobis comparavimus ab ipsa infantia, concipere omnino non liceret. 

Ceterum simplicium, & inextensorum notionem non ego primus in Physicam 
aiiis quoque ad- induco. Eorum ideam habuerunt veteres post Zenonem, & Leibnitiani monades suas & 
"rastare "hanc s i m P^ ces utique volunt, & inextensas ; ego cum ipsorum punctorum contiguitatem auferam, 
eorum theoriam. & distantias velim inter duo quaelibet materiae puncta, maximum evito scopulum, in quern 
utrique incurrunt, dum ex ejusmodi indivisibilibus, & inextensis continuum extensum 
componunt. Atque ibi quidem in eo videntur mini peccare utrique, quod cum simplicitate, 
& inextensione, quam iis elementis tribuunt, commiscent ideam illam imperfectam, quam 
sibi compararunt per sensus, globuli cujusdam rotundi, qui binas habeat superficies a se 
distinctas, utcumque interrogati, an id ipsum faciant, omnino sint negaturi. Neque enim 
aliter possent ejusmodi simplicibus inextensis implere spatium, nisi concipiendo unum 
elementum in medio duorum ab altero contactum ad dexteram, ab altero ad laevam, quin 
ea extrema se contingant; in quo, praeter contiguitatem indivisibilium, & inextensorum 
impossibilem, uti supra demonstravimus, quam tamen coguntur admittere, si rem altius 
perpenderint ; videbunt sane, se ibi illam ipsam globuli inter duos globules inter jacentis 
ideam admiscere. 

impugnatur con- 139. Nee ad indivisibilitatem, & inextensionem elementorum conjungendas cum 

formats ^b^inex- con t inua extensione massarum ab iis compositarum prosunt ea, quae nonnulli ex Leibniti- 

tensis petita ab anorum familia proferunt, de quibus egi in una adnotatiuncula adjecta num. 13. dissertationis 
impenetrabiiitate. j) g Mater its Divisibilitate, ? Principiis Corporum, ex qua, quae eo pertinent, hue libet 
transferre. Sic autem habet : Qui dicunt, monades non compenetrari, quia natura sua 
impenetrabiles sunt, ii difficultatem nequaquam amovenf ; nam si e? natura sua impenetrable s 
sunt, y continuum debent componere, adeoque contigua esse ; compenetrabuntur simul, W non 
compenetrabuntur, quod ad absurdum deducit, W ejusmodi entium impossibilitatem evincit. 
Ex omnimodfs inextensionis, & contiguitatis notione evincitur, compenetrari debere argumento 
contra Zenonistas institute per tot stecula, if cui nunquam satis responsum est. Ex natura, 
qua in [63] iis supponitur, ipsa compenetratio excluditur, adeoque habetur contradictio, & 

inductionem a 140'. Sunt alii, quibus videri poterit, contra haec ipsa puncta indivisibilia, & inextensa 

sensibihbus com- ,1 ., T . . , ^ . . . . r . . r . K . 

positis, & extensis adniberi posse mductionis prmcipmm, a quo contmuitatis legem, & alias propnetates 
haud vaiere contra derivavimus supra, quae nos ad haec indivisibilia, & inextensa puncta deduxerunt. Videmus 

puncta simplicia, & t* *. . . ... ... ,. . .. ... 

inextensa. enim in matena omni, quae se uspiam nostns objiciat sensibus, extensionem, divisibihtatem, 

partes ; quamobrem hanc ipsam proprietatem debemus transferre ad elementa etiam per 
inductionis principium. Ita ii : at hanc difficultatem jam superius praeoccupavimus, ubi 
egimus de inductionis principio. Pendet ea proprietas a ratione sensibilis, & aggregati, cum 
nimirum sub sensus nostros ne composita quidem, quorum moles nimis exigua sit, cadere 
possint. Hinc divisibilitatis, & extensionis proprietas ejusmodi est ; ut ejus defectus, si 
habeatur alicubi is casus, ex ipsa earum natura, & sensuum nostrorum constitutione non 
possit cadere sub sensus ipsos, atque idcirco ad ejusmodi proprietates argumentum desumptum 
ab inductione nequaquam pertingit, ut nee ad sensibilitatem extenditur. 

Per ipsam etiam 141. Sed etiam si extenderetur, esset adhuc nostrae Theoriae causa multo melior in eo, 

tensT^Hn^uctioms q 110 ^ circa, extensionem, & compositionem partium negativa sit. Nam eo ipso, quod 

habitam ipsum ex- continuitate admissa, continuitas elementorum legitima ratiocinatione excludatur, excludi 

omnino debet absolute ; ubi quidem illud accidit, quod a Metaphysicis, & Geometris 

nonnullis animadversum est jam diu, licere aliquando demonstrare propositionem ex 


From this it comes about that two particles of matter cannot be contiguous ; for thereupon 
they would recoil from one another owing to that repulsive force, & a particle composed 
of them would at once be broken up. Thus, the primary elements of matter cannot be 
composed of contiguous parts, but must be perfectly indivisible & simple ; and also on 
account of the induction from separability & the distinction between those that occupy 
different divisible parts of space, they must be perfectly non-extended as well. The idea 
acquired by reflection only yields the one result, namely, that through it we may form 
a clear conception of that which reasoning of this kind proves to be existent in Nature ; 
of which, without reflection, using only the equipment that we have got together for 
ourselves by means of the senses from our infancy, we could not have formed any 

138. Besides, I was not the first to introduce the notion of simple non-extended points Simple and 
into physics. The ancients from the time of Zeno had an idea of them, & the followers are^admitt 
of Leibniz indeed suppose that their monads are simple & non-extended. I, since I do others as well ; but 
not admit the contiguity of the points themselves, but suppose that any two points of ^m is "the 7 best." 
matter are separated from one another, avoid a mighty rock, upon which both these others 
come to grief, whilst they build up an extended continuum from indivisible & non-extended 
things of this sort. Both seem to me to have erred in doing so, because they have mixed 
up with the simplicity & non-extension that they attribute to the elements that imperfect 
idea of a sort of round globule having two surfaces distinct from one another, an idea they 
have acquired through the senses ; although, if they were asked if they had made this 
supposition, they would deny that they had done so. For in no other way can they fill up 
space with indivisible and non-extended things of this sort, unless by imagining that one 
element between two others is touched by one of them on the right & by the other on 
the left. If such is their idea, in addition to contiguity of indivisible & non-extended 
things (which is impossible, as I proved above, but which they are forced to admit if they 
consider the matter more carefully) ; in addition to this, I say, they will surely see that they 
have introduced into their reasoning that very idea of the two little spheres lying between 
two others. 

I3Q. Those arguments that some of the Leibnitian circle put forward are of no use The deduction from 

, i ~ r T -i ! o r i i -i. impenetrability of 

for the purpose of connecting indivisibility & non-extension of the elements with continuous a conciliation of 

extension of the masses formed from them. I discussed the arguments in question in extension ^j 1 ^ 

a short note appended to Art. 13 of the dissertation De Materies Divisibilitate and extendeTthings. 

Principiis Corporum ; & I may here quote from that dissertation those things that concern 

us now. These are the words : Those, who say that monads cannot be corn-penetrated, because 

they are by nature impenetrable, by no means remove the difficulty. For, if they are both by 

nature impenetrable, & also at the same time have to make up a continuum, i.e., have to be 

contiguous, then at one & the same time they are compenetrated & they are not compenetrated ; 

y this leads to an absurdity \3 proves the impossibility of entities of this sort. For, from the 

idea of non-extension of any sort, & of contiguity, it is proved by an argument instituted 

against the Zenonists many centuries ago that there is bound to be compenetration ; & -this 

argument has never been satisfactorily answered. From the nature that is ascribed to them, 

this compenetration is excluded. Thus there is a contradiction 13 an absurdity. 

140. There are others, who will think that it is possible to employ, for the purpose induction derived 
of opposing the idea of these indivisible & non-extended points, the principle of induction, ^T'senslSf 3 <m*- 
by which we derived the Law of Continuity & other properties, which have led us to pound, and ex- 
these indivisible & non-extended points. For we perceive (so they say) in all matter, a vau ed for r the f pur 
that falls under our notice in any way, extension, divisibility & parts. Hence we must pose of opposing 
transfer this property to the elements also by the principle of induction. Such is their 

argument. But we have already discussed this difficulty, when we dealt with the principle 
of induction. The property in question depends on a reasoning concerned with a sensible 
body, & one that is an aggregate ; for, in fact, not even a. composite body can come within 
the scope of our senses, if its mass is over-small. Hence the property of divisibility & 
extension is such that the absence of this property (if this case ever comes about), from 
the very nature of divisibility & extension, & from the constitution of our senses, cannot 
fall within the scope of those senses. Therefore an argument derived from induction will 
not apply to properties of this kind in any way, inasmuch as the extension does not reach 
the point necessary for sensibility. 

141. But even if this point is reached, there would only be all the more reason for our Extension 
Theory from the fact that it denies extension and composition by parts. For, from the very exclusion of 
fact that, if continuity be admitted, continuity of the elements is excluded by legitimate exte ^ s e io 
argument, it follows that continuity ought to be absolutely excluded in all cases. For in d u Ctio n. 
that case we get an instance of the argument that has been observed by metaphysicists 

and some geometers for a very long time, namely, that a proposition may sometimes be 


assumpta veritate contradictoriae propositionis ; cum enim ambae simul verae esse non 
possint, si ab altera inferatur altera, hanc posteriorem veram esse necesse est. Sic nimirum, 
quoniam a continuitate generaliter assumpta defectus continuitatis consequitur in materiae 
elementis, & in extensione, defectum hunc haberi vel inde eruitur : nee oberit 
quidquam principium inductionis physicae, quod utique non est demonstrativum, nee vim 
habet, nisi ubi aliunde non demonstretur, casum ilium, quern inde colligere possumus, 
improbabilem esse tantummodo, adhuc tamen haberi, uti aliquando sunt & falsa veris 

Cujusmodi con- 142. Atque hie quidem, ubi de continuitate seipsam excludente mentio injecta est, 

TheoiSadrnittatur n tandum & illud, continuitatis legem a me admitti, & probari pro quantitatibus, quae 
quid sit spatium, magnitudinem mutent, quas nimirum ab una magnitudine ad aliam censeo abire non posse, 
& tempus. n j g - transean t per intermedias, quod elementorum materiae, quse magnitudinem nee mutant, 

nee ullam habent variabilem, continuitatem non inducit, sed argumento superius facto 
penitus summovet. Quin etiam ego quidem continuum nullum agnosco coexistens, uti & 
supra monui ; nam nee spatium reale mihi est ullum continuum, sed [64] imaginarium 
tantummodo, de quo, uti & de tempore, quae in hac mea Theoria sentiam, satis luculenter 
exposui in Supplementis ad librum i. Stayanae Philosophise (*). Censeo nimirum quodvis 
materiae punctum, habere binos reales existendi modos, alterum localem, alterum tem- 
porarium, qui num appellari debeant res, an tantummodo modi rei, ejusmodi litem, quam 
arbitror esse tantum de nomine, nihil omnino euro. Illos modos debere admitti, ibi ego 
quidem positive demonstro : eos natura sua immobiles esse, censeo ita, ut idcirco ejusmodi 
existendi modi per se inducant relationes prioris, & posterioris in tempore, ulterioris, vel 
citerioris in loco, ac distantiae cujusdam deter minatae, & in spatio determinatae positionis 
etiam, qui modi, vel eorum alter, necessario mutari debeant, si distantia, vel etiam in spatio 
sola mutetur positio. Pro quovis autem modo pertinente ad quodvis punctum, penes 
omnes infinites modos possibiles pertinentes ad quodvis aliud, mihi est unus, qui cum eo 
inducat in tempore relationem coexistentiae ita, ut existentiam habere uterque non possit, 
quin simul habeant, & coexistant ; in spatio vero, si existunt simul, inducant relationem 
compenetrationis, reliquis omnibus inducentibus relationem distantiae temporarise, vel 
localis, ut & positionis cujusdam localis determinatae. Quoniam autem puncta materiae 
existentia habent semper aliquam a se invicem distantiam, & numero finita sunt ; finitus est 
semper etiam localium modorum coexistentium numerus, nee ullum reale continuum 
efformat. Spatium vero imaginarium est mihi possibilitas omnium modorum localium 
confuse cognita, quos simul per cognitionem praecisivam concipimus, licet simul omnes 
existere non possint, ubi cum nulli sint modi ita sibi proximi, vel remoti, ut alii viciniores, 
vel remotiores haberi non possint, nulla distantia inter possibiles habetur, sive minima 
omnium, sive maxima. Dum animum abstrahimus ab actuali existentia, & in possibilium 
serie finitis in infinitum constante terminis mente secludimus tarn minimae, quam maximae 
distantiae limitem, ideam nobis efformamus continuitatis, & infinitatis in spatio, in quo 
idem spatii punctum appello possibilitatem omnium modorum localium, sive, quod idem 
est, realium localium punctorum pertinentium ad omnia materiae puncta, quae si existerent, 
compenetrationis relationem inducerent, ut eodem pacto idem nomino momentum tem- 
poris temporarios modos omnes, qui relationem inducunt coexistentiae. Sed de utroque 
plura in illis dissertatiunculis, in quibus & analogiam persequor spatii, ac temporis 

Ubi habeat con- [65] 143. Continuitatem igitur agnosco in motu tantummodo, quod est successivum 
u i bi lit aff e e 1 ctet Na t Ura ^ u ^' non coexistens, & in eo itidem solo, vel ex eo solo in corporeis saltern entibus legem 
continuitatis admitto. Atque hinc patebit clarius illud etiam, quod superius innui, 
Naturam ubique continuitatis legem vel accurate observare, vel affectare saltern. ^ Servat in 
motibus, & distantiis, affectat in aliis casibus multis, quibus continuity, uti etiam supra 
definivimus, nequaquam convenit, & in aliis quibusdam, in quibus haberi omnino non pptest 
continuitas, quae primo aspectu sese nobis objicit res non aliquanto intimius inspectantibus, 
ac perpendentibus : ex. gr. quando Sol oritur supra horizontem, si concipiamus Solis discum 

(h) Binte dissertatiunculis, qua hue pertinent, inde excerptte habentur hie Supplementorum I, 13 2, quarum mentio 
facta est etiam superius num. 66, W 86. 



proved by assuming the truth of the contradictory proposition. For since both propositions 
cannot be true at the same time, if from one of them the other can be inferred, then the latter 
of necessity must be the true one. Thus, for instance, because it follows, from the 
assumption of continuity in general, that there is an absence of continuity in the elements 
of matter, & also in the case of extension, we come to the conclusion that there is this 
absence. Nor will any principle of physical induction be prejudicial to the argument, 
where the induction is not one that can be proved in every case ; neither will it have any 
validity, except in the case where it cannot be proved in other ways that the conclusion 
that we can come to from the argument is highly improbable but yet is to be held as 
true ; for indeed sometimes things that are false are more plausible than the true facts. 

142. Now, in this connection, whilst incidental mention has been made of the exclusion xhe sort of 

of continuity, it should be observed that the Law of Continuity is admitted by me, & tinuum that is 
proved for those quantities that change their magnitude, but which indeed I consider Th^r^fthe^ature 
cannot pass from one magnitude to another without going through intermediate stages ; of s P a ce and time, 
but that this does not lead to continuity in the case of the elements of matter, which neither 
change their magnitude nor have anything variable about them ; on the contrary it proves 
quite the opposite, as the argument given above shows. Moreover, I recognize no co- 
existing continuum, as I have already mentioned ; for, in my opinion, space is not any 
real continuum, but only an imaginary one ; & what I think about this, and about time 
as well, as far as this Theory is concerned, has been expounded clearly enough in the 
supplements to the first book of Stay's Philosophy. (A) For instance, I consider that any 
point of matter has two modes of existence, the one local and the other temporal ; I do 
not take the trouble to argue the point as to whether these ought to be called things, or 
merely modes pertaining to a thing, as I consider that this is merely a question of terminology. 
That it is necessary that these modes be admitted, I prove rigorously in the supplements 
mentioned above. I consider also that they are by their very nature incapable of being 
displaced ; so that, of themselves, such modes of existence lead to the relations of before 
& after as regards time, far & near as regards space, & also of a given distance & 
a given position in space. These modes, or one of them, must of necessity be changed, 
if the distance, or even if only the position in space is altered. Moreover, for any one 
mode belonging to any point, taken in conjunction with all the infinite number of possible 
modes pertaining to any other point, there is in my opinion one which, taken in conjunction 
with the first mode, leads as far as time is concerned to a relation of coexistence ; so that 
both cannot have existence unless they have it simultaneously, i.e., they coexist. But, 
as far as space is concerned, if they exist simultaneously, the conjunction leads to a relation 
of compenetration. All the others lead to a relation of temporal or of local distance, as 
also of a given local position. Now since existent points of matter always have some distance 
between them, & are finite in number, the number of local modes of existence is also 
always finite ; & from this finite number we cannot form any sort of real continuum. 
But I have an ill-defined idea of an imaginary space as a possibility of all local modes, which 
are precisely conceived as existing simultaneously, although they cannot all exist simul- 
taneously. In this space, since there are not modes so near to one another that there 
cannot be others nearer, or so far separated that there cannot be others more so, there 
cannot therefore be a distance that is either the greatest or the least of all, amongst those 
that are possible. So long as we keep the mind free from the idea of actual existence &, in 
a series of possibles consisting of an indefinite number of finite terms, we mentally exclude 
the limit both of least & greatest distance, we form for ourselves a conception of continuity 
& infinity in space. In this, I define the same point of space to be the possibility of all 
local modes, or what comes to the same thing, of real local points pertaining to all points 
of matter, which, if they existed, would lead to a relation of compenetration ; just as I 
define the same instant of time as all temporal modes, which lead to a relation of coexistence. 
But there is a fuller treatment of both these subjects in the notes referred to ; & in them 
I investigate further the manifold analogy between space & time. 

143. Hence I acknowledge continuity in motion only, which is something successive where there is con- 

i TJ . . . . , ' , f . V . . tmuity in Nature ; 

and not co-existent ; & also in it alone, or because or it alone, in corporeal entities at any W here Nature does 
rate, lies my reason for admitting the Law of Continuity. From this it will be all the no more than at- 
more clear that, as I remarked above, Nature accurately observes the Law of Continuity, j t eml 
or at least tries to do so. Nature observes it in motions & in distance, & tries to in many 
other cases, with which continuity, as we have defined it above, is in no wise in agree- 
ment ; also in certain other cases, in which continuity cannot be completely obtained. This 
continuity does not present itself to us at first sight, unless we consider the subjects somewhat 
more deeply & study them closely. For instance, when the sun rises above the horizon, 

(h) The two notes, which refer to this matter, have been quoted in this work as supplements IS- II : these have 
been already referred to in Arts. 66 & 86 above. 


ut continuum, & horizontem ut planum quoddam ; ascensus Solis fit per omnes magnitudines 
ita, ut a primo ad postremum punctum & segmenta Solaris disci, & chordae segmentorum 
crescant transeundo per omnes intermedias magnitudines. At Sol quidem in mea Theoria 
non est aliquid continuum, sed est aggregatum punctorum a se invicem distantium, quorum 
alia supra illud imaginarium planum ascendunt post alia, intervallo aliquo temporis inter- 
posito semper. Hinc accurata ilia continuitas huic casui non convenit, & habetur tantummodo 
in distantiis punctorum singulorum componentium earn massam ab illo imaginario piano. 
Natura tamen etiam hie continuitatem quandam affectat, cum nimirum ilia punctula ita 
sibi sint invicem proxima, & ita ubique dispersa, ac disposita, ut apparens quaedam ibi etiam 
continuitas habeatur, ac in ipsa distributione, a qua densitas pendet, ingentes repentini 
saltus non riant. 

Exempla continu- 144. Innumera ejus rei exempla liceret proferre, in quibus eodem pacto res pergit. 

it at is apparent gj c j n fl uv i orum alveis, in frondium flexibus, in ipsis salium, & crystallorum, ac aliorum 

tantum : unde ea ..... . ,., . *., 

ortum ducat. corporum angulis, in ipsis cuspidibus unguium, quae acutissimae in quibusdam ammalibus 
apparent nudo oculo ; si microscopio adhibito inspiciantur ; nusquam cuspis abrupta 
prorsus, nusquam omnino cuspidatus apparet angulus, sed ubique flexus quidam, qui 
curvaturam habeat aliquam, & ad continuitatem videatur accedere. In omnibus tamen iis 
casibus vera continuitas in mea Theoria habetur nusquam ; cum omnia ejusmodi corpora 
constent indivisibilibus, & a se distantibus punctis, quse continuam superficiem non efformant, 
& in quibus, si quaevis tria puncta per rectas lineas conjuncta intelligantur ; triangulum 
habebitur utique cum angulis cuspidatis. Sed a motuum, & virium continuitate accurata 
etiam ejusmodi proximam continuitatem massarum oriri censeo, & a casuum possibilium 
multitudine inter se collata, quod ipsum innuisse sit satis. 

Motuum omnium 145- Atque hinc fiet manifestum, quid respondendum ad casus quosdam, qui eo 

continuitas in -pertinent, & in quibus violari quis crederet F661 continuitatis legem. Quando piano aliquo 

line is continuis r .*. f r . n n 

nusquam inter- speculo lux excipitur, pars relrmgitur, pars renectitur : in renexione, & retractione, uti earn 
ruptis, aut mutatis. o li m creditum est fieri, & etiamnum a nonnullis creditur, per impulsionem nimirum, & 
incursum immediatum, fieret violatio quaedam continui motus mutata linea recta in aliam ; 
sed jam hoc Newtonus advertit, & ejusmodi saltum abstulit, explicando ea phenomena per 
vires in aliqua distantia agentes, quibus fit, ut quaevis particula luminis motum incurvet 
paullatim in accessu ad superficiem re flectentem, vel refringentem ; unde accessuum, & 
recessuum lex, velocitas, directionum flexus, omnia juxta continuitatis legem mutantur. 
Quin in mea Theoria non in aliqua vicinia tantum incipit flexus ille, sed quodvis materiae 
punctum a Mundi initio unicam quandam continuam descripsit orbitam, pendentem a 
continua ilia virium lege, quam exprimit figura I , quae ad distantias quascunque protenditur ; 
quam quidem lineae continuitatem nee liberae turbant animarum vires, quas itidem non nisi 
juxta continuitatis legem exerceri a nobis arbitror ; unde fit, ut quemadmodum omnem 
accuratam quietem, ita omnem accurate rectilineum motum, omnem accurate circularem, 
ellipticum, parabolicum excludam ; quod tamen aliis quoque sententiis omnibus commune 
esse debet ; cum admodum facile sit demonstrare, ubique esse perturbationem quandam, 
& mutationum causas, quae non permittant ejusmodi linearum nobis ita simplicium accuratas 
orbitas in motibus. 

Apparens saltus in 146. Et quidem ut in iis omnibus, & aliis ejusmodi Natura semper in mea Theoria 

diffusione reflexi, accuratissimam continuitatem observat, ita & hie in reflexionibus, ac refractionibus luminis. 

ac refracti luminis. . ,. , . ..'..,. , , , '. . , 

At est ahud ea in re, in quo continuitatis violatio quaedam haben videatur, quam, qui rem 
altius perpendat, credet primo quidem, servari itidem accurate a Natura, turn ulterius 
progressus, inveniet affectari tantummodo, non servari. Id autem est ipsa luminis diffusio, 
atque densitas. Videtur prima fronte discindi radius in duos, qui hiatu quodam intermedio 
a se invicem divellantur velut per saltum, alia parte reflexa, ali refracta, sine ullo intermedio 
flexu cujuspiam. Alius itidem videtur admitti ibidem saltus quidam : si enim radius 
integer excipiatur prismate ita, ut una pars reflectatur, alia transmittatur, & prodeat etiam 
e secunda superficie, turn ipsum prisma sensim convertatur ; ubi ad certum devenitur in 
conversione angulum, lux, quae datam habet refrangibilitatem, jam non egreditur, sed 
reflectitur in totum ; ubi itidem videtur fieri transitus a prioribus angulis cum superficie 
semper minoribus, sed jacentibus ultra ipsam, ad angulum reflexionis aequalem angulo 


if we think of the Sun's disk as being continuous, & the horizon as a certain plane ; then 
the rising of the Sun is made through all magnitudes in such a way that, from the first to 
the last point, both the segments of the solar disk & the chords of the segments increase by 
passing through all intermediate magnitudes. But, in my Theory, the Sun is not something 
continuous, but is an aggregate of points separate from one another, which rise, one after 
the other, above that imaginary plane, with some interval of time between them in all 
cases. Hence accurate continuity does not fit this case, & it is only observed in the case 
of the distances from the imaginary plane of the single points that compose the mass of the 
Sun. Yet Nature, even here, tries to maintain a sort of continuity ; for instance, the 
little points are so very near to one another, & so evenly spread & placed that, even in 
this case, we have a certain apparent continuity, and even in this distribution, on which 
the density depends, there do not occur any very great sudden changes. 

144. Innumerable examples of this apparent continuity could be brought forward, in Examples of con- 
which the matter comes about in the same manner. Thus, in the channels of rivers, the ^"reiy apparent' 3 
bends in foliage, the angles in salts, crystals and other bodies, in the tips of the claws that its origin, 
appear to the naked eye to be very sharp in the case of certain animals ; if a microscope 

were used to examine them, in no case would the point appear to be quite abrupt, or the 
angle altogether sharp, but in every case somewhat rounded, & so possessing a definite 
curvature & apparently approximating to continuity. Nevertheless in all these cases 
there is nowhere true continuity according to my Theory ; for all bodies of this kind are 
composed of points that are indivisible & separated from one another ; & these cannot 
form a continuous surface ; & with them, if any three points are supposed to be joined 
by straight lines, then a triangle will result that in every case has three sharp angles. But 
I consider that from the accurate continuity of motions & forces a very close approximation 
of this kind arises also in the case of masses ; &, if the great number of possible cases are 
compared with one another, it is sufficient for me to have just pointed it out. 

145. Hence it becomes evident how we are to refute certain cases, relating to this The . continuity of 
matter, in which it might be considered that the Law of Continuity was violated. When uous lines ""is 
light falls upon a plane mirror, part is refracted & part is reflected. In reflection & nowhere inter- 
refraction, according to the idea held in olden times, & even now credited by some people, rup e 
namely, that it took place by means of impulse & immediate collision, there would be 

a breach of continuous motion through one straight line being suddenly changed for 
another. But already Newton has discussed this point, & has removed any sudden change 
of this sort, by explaining the phenomena by means of forces acting at a distance ; with 
these it comes about that any particle of light will have its path bent little by little as it 
approaches a reflecting or refracting surface. Hence, the law of approach and recession, 
the velocity, the alteration of direction, all change in accordance with the Law of Continuity. 
Nay indeed, in my Theory, this alteration of direction does not only begin in the immediate 
neighbourhood, but any point of matter from the time that the world began has described 
a single continuous orbit, depending on the continuous law of forces, represented in Fig. i, 
a law that extends to all distances whatever. I also consider that this continuity of path 
is undisturbed by any voluntary mental forces, which also cannot be exerted by us except 
in accordance with the Law of Continuity. Hence it comes about that, just as I exclude 
all idea of absolute rest, so I exclude all accurately rectilinear, circular, elliptic, or parabolic 
motions. This too ought to be the general opinion of all others ; for it is quite easy to show 
that there is everywhere some perturbation, & reasons for alteration, which do not allow 
us to have accurate paths along such simple lines for our motions. 

146. Just as in all the cases I have mentioned, & in others like them, Nature always Apparent discon- 

'.-,,, J .. i i i i i tinuity in diffusion 

in my Theory observes the most accurate continuity, so also is this done here in the case O f ren ected and re- 

of the reflection and refraction of light. But there is another thing in this connection, fracted light. 

in which there seems to be a breach of continuity ; & anyone who considers the matter 

fairly deeply, will think at first that Nature has observed accurate continuity, but on further 

consideration will find that Nature has only endeavoured to do so, & has not actually 

observed it ; that is to say, in the diffusion of light, & its density. At first sight the ray 

seems to be divided into two parts, which leave a gap between them & diverge from one 

another as it were suddenly, the one part being reflected & the other part refracted 

without any intermediate bending of the path. It also seems that another sudden change 

must be admitted ; for suppose that a beam of light falls upon a prism, & part of it is 

reflected & the rest is transmitted & issues from the second surface, and that then the 

prism is gradually rotated ; when a certain angle of rotation is reached, light, having 

a given refrangibility, is no longer transmitted, but is totally reflected. Here also it 

seems that there is a sudden transition from the first case in which the angles made^with 

the surface by the issuing rays are always less than the angle of incidence, & lie on 

the far side of the surface, to the latter case in which the angles of reflection are equal to 


incidentiae, & jacentem citra, sine ulla reflexione in angulis intermediis minoribus ab ipsa 
superficie ad ejusmodi finitum angulum. 

Apparens concili- 14.7. Huic cuidam velut laesioni continuitatis videtur responderi posse per illam lucem 

Unuitafe pel radios q ua3 reflectitur, vel refrin-[67]-gitur irregulariter in quibusvis angulis. Jam olim enim 
irregulariter disper- observatum est illud, ubi lucis radius reflectitur, non reflecti totum ita, ut angulus 
reflexionis aequetur angulo incidentiae, sed partem dispergi quaquaversus ; quam ob causam 
si Solis radius in partem quandam speculi incurrat, quicunque est in conclavi, videt, qui sit 
ille locus, in quern incurrit radius, quod utique non fieret, nisi e solaribus illis directis radiis 
etiam ad oculum ipsius radii devenirent, egressi in omnibus iis directionibus, quae ad omnes 
oculi positiones tendunt ; licet ibi quidem satis intensum lumen non appareat, nisi in 
directione faciente angulum reflexionis aequalem incidentiae, in qua resilit maxima luminis 
pars. Et quidem hisce radiis redeuntibus in angulis hisce inaequalibus egregie utitur 
Newtonus in fine Opticae ad explicandos colores laminarum crassarum : & eadem irregularis 
dispersio in omnes plagas ad sensum habetur in tenui parte, sed tamen in aliqua, radii 
refracti. Hinc inter vividum ilium reflexum radium, & refractum, habetur intermedia 
omnis ejusmodi radiorum series in omnibus iis intermediis angulis prodeuntium, & sic etiam 
ubi transitur a refractione ad reflexionem in totum, videtur per hosce intermedios angulos 
res posse fieri citissimo transitu per ipsos, atque idcirco illaesa perseverare continuitas. 

Cur ea apparens 148. Verum si adhuc altius perpendatur res ; patebit in ilia intermedia serie non haberi 

dSitio 1 pe^contiii- accuratam continuitatem, sed apparentem quandam, quam Natura affectat, non accurate 

ujtatem yiae cujus- servat illaesam. Nam lumen in mea Theoria non est corpus quoddam continuum, quod 

vis puncti diffundatur continue per illud omne spatium, sed est aggregatum punctorum a se invicem 

disjunctorum, atque distantium, quorum quodlibet suam percurrit viam disjunctam a 

proximi via per aliquod intervallum. Continuitas servatur accuratissime in singulorum 

punctorum viis, non in diffusione substantiae non continuae, & quo pacto ea in omnibus iis 

motibus servetur, & mutetur, mutata inclinatione incidentiae, via a singulis punctis descripta 

sine saltu, satis luculenter exposui in secunda parte meae dissertationis De Lumine a num. 98. 

Sed haec ad applicationem jam pertinent Theoriae ad Physicam. 

QUO pacto servetur 149. Haud multum absimiles sunt alii quidam casus, in quibus singula continuitatem 

bu^dam^casibusTui observant, non aggregatum utique non continuum, sed partibus disjunctis constans. 
quibus videtur tedi. Hujusmodi est ex. gr. altitude cujusdam domus, quae aedificatur de novo, cui cum series 
nova adjungitur lapidum determinatae cujusdam altitudinis, per illam additionem repente 
videtur crescere altitude domus, sine transitu per altitudines intermedias : & si dicatur id 
non esse Naturae opus, sed artis ; potest difficultas transferri facile ad Naturae opera, ut ubi 
diversa inducuntur glaciei strata, vel in aliis incrustationibus, ac in iis omnibus casibus, in 
quibus incrementum fit per externam applicationem partium, ubi accessiones finitae videntur 
acquiri simul totae sine [68] transitu per intermedias magnitudines. In iis casibus 
continuitas servatur in motu singularum partium, quae accedunt. Illae per lineam quandam 
continuam, & continua velocitatis mutatione accedunt ad locum sibi deditum : quin immo 
etiam posteaquam eo advenerunt, pergunt adhuc moveri, & nunquam habent quietem nee 
absolutam, nee respectivam respectu aliarum partium, licet jam in respectiva positione 
sensibilem mutationem non subeant : parent nimirum adhuc viribus omnibus, quae 
respondent omnibus materiae punctis utcunque distantibus, & actio proximarum partium, 
quae novam adhaesionem parit, est continuatio actionis, quam multo minorem exercebant, 
cum essent procul. Hoc autem, quod pertineant ad illam domum, vel massam, est aliquid 
non in se determinatum, quod momento quodam determinato fiat, in quo saltus habeatur, 
sed ab aestimatione quadam pendet nostrorum sensuum satis crassa ; ut licet perpetuo 
accedant illae partes, & pergant perpetuo mutare positionem respectu ipsius massae ; turn 
incipiant censeri ut pertinentes ad illam domum, vel massam : cum desinit respectiva 
mutatio esse sensibilis, quae sensibilitatis cessatio fit ipsa etiam quodammodo per gradus 
omnes, & continue aliquo tempore, non vero per saltum. 

Generate responsio ISO- Hinc distinctius ibi licebit difHcultatem omnem amovere dicendo, non servari 

de emta. 3 similes m " mutationem continuam in magnitudinibus earum rerum, quae continuae non sunt, & 

magnitudinem non habent continuam, sed sunt aggregata rerum disjunctarum ; vel in iis 

rebus, quae a nobis ita censentur aliquod totum constituere, ut magnitudinem aggregati non 


the angles of incidence & lie on the near side of the surface, without any reflection for 
rays at intermediate angles with the surface less than a certain definite angle. 

147. It seems that an explanation of this apparent breach of continuity can be given Apparent recontiii- 
by means of light that is reflected or refracted irregularly at all sorts of angles. For long ago of Continuity 6 effect 
it was observed that, when a ray of light is reflected, it is not reflected entirely in such a * ed fa y means of 
manner that the angle of reflection is equal to the angle of incidence, but that a part of it 

is dispersed in all directions. For this reason, if a ray of light from the Sun falls upon some 
part of a mirror, anybody who is in the room sees where the ray strikes the mirror ; & 
this certainly would not be the case, unless some of the solar rays reached his eye directly 
issuing from the mirror in all those directions that reach to all positions that the eye might 
be in. Nevertheless, in this case the light does not appear to be of much intensity, unless 
the eye is in the position facing the angle of reflection equal to the angle of incidence, along 
which the greatest part of the light rebounds. Newton indeed employed in a brilliant 
way these rays that issue at irregular angles at the end of his Optics to explain the colours 
of solid laminae. The same irregular dispersion in all directions takes place as far as can 
be observed in a small part, but yet in some part, of the refracted ray. Hence, in between 
the intense reflected & refracted rays, we have a whole series of intermediate rays of this sort 
issuing at all intermediate angles. Thus, when the transition is effected from refraction 
to total reflection, it seems that it can be done through these intermediate angles by an 
extremely rapid transition through them, & therefore continuity remains unimpaired. 

148. But if we inquire into the matter yet more carefully, it will be evident that in Why this is only an 
that intermediate series there is no accurate continuity, but only an apparent continuity ; atimT* the^true 
& this Nature tries to maintain, but does not accurately observe it unimpaired. For, reconciliation is 
in my Theory, light is not some continuous body, which is continuously diffused through t!nurty h of ^ath^or 
all the space it occupies ; but it is an aggregate of points unconnected with & separated any point of light, 
from one another ; & of these points, any one pursues its own path, & this path is separated 

from the path of the next point to it by a definite interval. Continuity is observed perfectly 
accurately for the paths of the several points, not in the diffusion of a substance that is 
not continuous ; & the manner in which continuity is preserved in all these motions, 
& the path described by the several points is altered without sudden change, when the angle 
of incidence is altered, I have set forth fairly clearly in the second part of my dissertation 
De Lumine, Art. 98. But in this work these matters belong to the application of the 
Theory to physics. 

140. There are certain cases, not greatly unlike those already given, in which each The manner in 

/ i _j i_ ' V j r which continuity 

part preserves continuity, but not so the whole, which is not continuous but composed ot is m aintained m 

separate parts. For an instance of this kind, take the height of a new house which is being certain cases in 

built ; as a fresh layer of stones of a given height is added to it, the height of the house ^ 

on account of that addition seems to increase suddenly without passing through intermediate 

heights. If it is said that that is not a work of Nature, but of art ; then the same difficulty 

can easily be transferred to works of Nature, as when different strata of ice are formed, or 

in other incrustations, and in all cases in which an increment is caused by the external 

application of parts, where finite additions seem to be acquired all at once without any 

passage through intermediate magnitudes. In these cases the continuity is preserved in 

the motions of the separate parts that are added. These reach the place allotted to them 

along some continuous line & with a continuous change of velocity. Further, after they 

have reached it, they still continue to move, & never have absolute rest ; no, nor even 

relative rest with respect to the other parts, although they do not now suffer a sensible 

change in their relative positions. Thus, they still submit to the action of all the forces 

that correspond to all points of matter at any distances whatever ; and the action of the 

parts nearest to them, which produces a new adhesion, is the continuation of the action 

that they exert to a far smaller extent when they are some distance away. Moreover, in 

the fact that they belong to that house or mass, there is something that is not determinate 

in itself, because it happens at a determinate instant in which the sudden change takes 

place ; but it depends on a somewhat rough assessment by our senses. So that, although 

these parts are continually being added, & continually go on changing their position 

with respect to the mass, they both begin to be thought of as belonging to that house or 

mass, & the relative change ceases to be sensible ; also this cessation of sensibility itself 

also takes place to some extent through all stages, and in some continuous interval of time, 

& not by a sudden jump. 

KO. From this consideration we may here in a clearer manner remove all difficulty 

-' * . , . - jrt,'U* simuar cd.b 

by saying that a continuous change is not maintained in the magnitudes ot those tmngs, derived from this, 
which are not themselves continuous, & do not possess continuous magnitude, but are 
aggregates of separate things. That is to say, in those things that are thus considered as 
forming a certain whole, in such a way that the magnitude of the aggregate is not determined 



determinent distantias inter eadem extrema, sed a nobis extrema ipsa assumantur jam alia, 
jam alia, quae censeantur incipere ad aggregatum pertinere, ubi ad quasdam distantias 
devenerint, quas ut ut in se juxta legem continuitatis mutatas, nos a reliquis divellimus per 
saltum, ut dicamus pertinere eas partes ad id aggregatum. Id accidit, ubi in objectis 
casibus accessiones partium novae fiunt, atque ibi nos in usu vocabuli saltum facimus ; ars, 
& Natura saltum utique habet nullum. 

Alii casus in quibus 151. Non idem contingit etiam, ubi plantas, vel animantia crescunt, succo se insinuante 

'uibus Ur ' hab'etur P er tubulos fibrarum, & procurrente, ubi & magnitude computata per distantias punctorum 
soium proxima, non maxime distantium transit per omnes intermedias ; cum nimirum ipse procursus fiat 
accurata contmm- p ef omnes intermedias distantias. At quoniam & ibi mutantur termini illi, qui distantias 
determinant, & nomen suscipiunt altitudinis ipsius plantas ; vera & accurata continuitas ne 
ibi quidem observatur, nisi tantummodo in motibus, & velocitatibus, ac distantiis singularum 
partium : quanquam ibi minus recedatur a continuitate accurata, quam in superioribus. In 
his autem, & in illis habetur ubique ilia alia continuitas quasdam apparens, & affectata 
tantummodo a Natura, quam intuemur etiam in progressu substantiarum, ut incipiendo ab 
inanima-[69]-tis corporibus progressu facto per vegetabilia, turn per quasdam fere 
semianimalia torpentia, ac demum animalia perfectiora magis, & perfectiora usque ad simios 
homini tarn similes. Quoniam & harum specierum, ac existentium individuorum in quavis 
specie numerus est finitus, vera continuitas haberi non potest, sed ordinatis omnibus in 
seriem quandam, inter binas quasque intermedias species hiatus debet esse aliquis necessario, 
qui continuitatem abrumpat. In omnibus iis casibus habentur discretas quasdam quantitates, 
non continues ; ut & in Arithmetica series ex. gr. naturalium numerorum non est continua, 
sed discreta ; & ut ibi series ad continuam reducitur tantummodo, si generaliter omnes 
intermedias fractiones concipiantur ; sic & in superiore exemplo quasdam velut continua 
series habebitur tantummodo ; si concipiantur omnes intermedias species possibiles. 


Conciusio pertinens 152. Hoc pacto excurrendo per plurimos justmodi casus, in quibus accipiuntur 

ad ea, quse veram, a g gre g ata rerum a se invicem certis intervallis distantium, & unum aliquid continuum non 

(X CcL, CJ1.13E cLttCCtcl" OO O iill** 

tam habent contin- constituentium, nusquam accurata occurret continuitatis lex, sed per quandam dispersionem 
quodammodo affectata, & vera continuitas habebitur tantummodo in motibus, & in iis, quas 
a motibus pendent, uti sunt distantiae, & vires determinatas a distantiis, & velocitates a 
viribus ortae ; quam ipsam ob causam ubi supra num. 39 inductionem pro lege continuitatis 
assumpsimus, exempla accepimus a motu potissimum, & ab iis, quae cum ipsis motibus 
connectuntur, ac ab iis pendent. 

153. Sed jam ad aliam difficultatem gradum faciam, quae non nullis negotium ingens 
3ito facessit, & obvia est etiam, contra hanc indivisibilium, & inextensorum punctorum Theoriam ; 
' & quod nimirum ea nullum habitura sint discrimen a spiritibus. Ajunt enim, si spiritus 
ejusmodi vires habeant, praestituros eadem phaenomena, tolli nimirum corpus, & omnem 
corporeae substantiae notionem sublata extensione continua, quae sit prascipua materias 
proprietas ita pertinens ad naturam ipsius ; ut vel nihil aliud materia sit, nisi substantia 
praedita extensione continua ; vel saltern idea corporis, & materiae haberi non ppssit ; nisi 
in ea includatur idea extensionis continuae. Multa hie quidem congeruntur simul, quae 
nexum aliquem inter se habent, quae hie seorsum evolvam singula. 

Difficultates petitae 
a discrimine debito 
inter materiam 

DifferrehKcpuncta 154. Inprimis falsum omnino est, nullum esse horum punctorum discrimen a spiritibus. 

fm^netrabUttlte^ Di scrimen potissimum materiae a spiritu situm est in hisce duobus, quod _ materia_ est 

nSitatem, a e - sensibilis, & incapax cogitationis, ac voluntatis, spiritus nostros sensus non afficit, & cogitare 

capadtatem cogit- p Otest) ac ve lle. Sensibilitas autem non ab extensione continua oritur, sed ab impene- 

trabilitate, qua fit, ut nostrorum organorum fibrae tendantur a corporibus, quae ipsis 

sistuntur, & motus ad cerebrum pro-[7o]-pagetur. Nam si extensa quidem essent corpora, 

sed impenetrabilitate carerent ; manu contrectata fibras non sisterent, nee motum ullum 

in iis progignerent, ac eadem radios non reflecterent, sed liberum intra se aditum luci 

prasberent. Porro hoc discrimen utrumque manere potest integrum, & manet inter mea 

indivisibilia hasc puncta, & spiritus. Ipsa impenetrabilitatem habent, & sensus nostros 

afficiunt, ob illud primum crus asymptoticum exhibens vim illam repulsivam primam ; 

spiritus autem, quos impenetrabilitate carere credimus, ejusmodi viribus itidem carent, & 

sensus nostros idcirco nequaquam afficiunt, nee oculis inspectantur, nee ^manibus palpari 

possunt. Deinde in meis hisce punctis ego nihil admitto aliud, nisi illam virium legem cum 

inertias vi conjunctam, adeoque ilia volo prorsus incapacia cogitationis, & voluntatis. 


I2 3 

by the distances between the same extremes all the time, but the extremes we take are 
different, one after another ; & these are considered to begin to belong to the aggregate 
when they attain to certain distances from it ; &, although in themselves changed in 
accordance with the Law of Continuity, we separate them from the rest in a discontinuous 
manner, by saying that these parts belong to the aggregate. This comes about, whenever 
in the cases under consideration fresh additions of parts take place ; & then we make a 
discontinuity in the use of a term ; art, as well as Nature, has no discontinuity. 

151. It is not the same thing however in the case of the growth of plants or animals, 
which is due to a life-principle insinuating itself into, & passing along the fine tubes of the 
fibres ; here the magnitude, calculated by means of the distance between the points furthest 
from one another, passes through all intermediate distances ; for the flow of the life-principle 
takes place indeed through all intermediate distances. But, since here also the extremes 
are changed, which determine the distances, & denominate the altitude of the plant ; 
not even in this case is really accurate continuity observed, except only in the motions & 
velocities and distances of the separate parts ; however there is here less departure from 
accurate continuity, than there was in the examples given above. In both there is indeed 
that kind of apparent continuity, which Nature does no more than try to maintain ; such 
as we also see in the series of substantial things, which starting from inanimate bodies, 
continues through vegetables, then through certain sluggish semianimals, & lastly, through 
animals more & more perfect, up to apes that are so like to man. Also, since the number 
of these species, & the number of existent individuals of any species, is finite, it is impossible 
to have true continuity ; but if they are all ordered in a series, between any two intermediate 
species there must necessarily be a gap ; & this will break the continuity. In all these 
cases we have certain discrete, & not continuous, quantities ; just as, for instance, the 
arithmetical series of the natural numbers is not continuous, but discrete. Also, just as the 
series is reduced to continuity only by mentally introducing in general all the intermediate 
fractions ; so also, in the example given above a sort of continuous series is obtained, if 
& only if all intermediate possible species are so included. 

152. In the same way, if we examine a large number of cases of the same kind, in which 
aggregates of things are taken, separated from one another by certain definite intervals, 
& not composing a single continuous whole, an accurate continuity law will never be 
met with, but only a sort of counterfeit depending on dispersion. True continuity will 
only be obtained in motions, & in those things that depend on motions, such as distances 
& forces determined by distances, & velocities derived from such forces. It was for 
this very reason that, when we adopted induction for the proof of the Law of Continuity 
in Art. 39 above, we took our examples mostly from motion, & from those things which 
are connected with motions & depend upon them. 

153. Now I will pass on to another objection, which some people have made a great 
to-do about, and which has also been raised in opposition to this Theory of indivisible & 
non-extended points ; namely, that there will be no difference between my points & 
spirits. For, they say that, if spirits were endowed with such forces, they would show the 
same phenomena as bodies, & that bodies & all idea of corporeal substance would be 
done away with by denying continuous extension ; for this is one of the chief properties of 
matter, so pertaining to Nature itself ; so that either matter is nothing else but substance 
endowed with continuous extension, or the idea of a body and of matter cannot be obtained 
without the inclusion of the idea of continuous extension. Here indeed there are many 
matters all jumbled together, which have no connection with one another ; these I will 
now separate & discuss individually. 

154. First of all it is altogether false that there is no difference between my points & 
spirits. The most important difference between matter & spirit lies in the two facts, 
that matter is sensible & incapable of thought, whilst spirit does not affect the senses, 
but can think or will. Moreover, sensibility does not arise from continuous extension, 
but from impenetrability, through which it comes about that the fibres of our organs are 
subjected to stress by bodies that are set against them & motions are thereby propagated 
to the brain. For if indeed bodies were extended, but lacked impenetrability, they would 
not resist the fibres of the hand when touched, nor produce in them any motion ; nor 
would they reflect light, but allow it an uninterrupted passage through themselves. 
Further, it is possible that each of these distinctions should hold good independently ; 
& they do so between these indivisible points of mine & spirits. My points have 
impenetrability & affect our senses, because of that first asymptotic branch representing that 
first repulsive force ; but spirits, which we suppose to lack impenetrability, lack also forces 
of this kind, and therefore can in no wise affect our senses, nor be examined by the eyes, 
nor be felt by the hands. Then, in these points of mine, I admit nothing else but the 
law of forces conjoined with the force of inertia ; & hence I intend them to be incapable 

Cases in which 
there is a breach of 
continuity ; others 
in which the con- 
tinuity is only very 
nearly, but not 
accurately, ob- 

Conclusion as re- 
gards those things 
that possess true 
continuity, and 
those that have a 
counterfeit continu- 

Objections derived 
from the distinc- 
tion that has to be 
made between 
matter & spirit. 

These points differ 
from spirits on 
account o f their 
impenet rability, 
their being sen- 
sible, & their inca- 
pacity for thought. 

I2 4 


Si possibilis sub- 

earn nee 

Quamobrem discrimen essentiae illud utrumque, quod inter corpus, & spiritum 
agnoscunt omnes, id & ego agnosco, nee vero id ab extensione, & compositione continua 
desumitur, sed ab iis, quae cum simplicitate, & inextensione aeque conjungi possunt, & 
cohaerere cum ipsis. 

155. At si substantiae capaces cogitationis & voluntatis haberent ejusmodi virium legem, 
an non eosdem praestarent effectus respectu nostrorum sensuum, quos ejusmodi puncta ? 
capax cogitationis ; Respondebo sane, me hie non quaerere, utrum impenetrabilitas, & sensibilitas, quae ab iis 
f nec v i r ibus pendent, conjungi possint cum facultate cogitandi, & volendi, quae quidem quaestio 
eodem redit, ac in communi sententia de impenetrabilitate extensorum, ac compositorum 
relata ad vim cogitandi, & volendi. Illud ajo, notionem, quam habemus partim ex 
observationibus tarn sensuum respectu corporurh, quam intimae conscientiae respectu 
spiritus, una cum reflexione, partim, & vero etiam circa spiritus potissimum, ex principiis 
immediate revelatis, vel connexis cum principiis revelatis, continere pro materia 
impenetrabilitatem, & sensibilitatem, una cum incapacitate cogitationis, & pro spiritu 
incapacitatem afHcicndi per impenetrabilitatem nostros sensus, & potentiam cogitandi, ac 
volendi, quorum priores illas ego etiam in meis punctis admitto, posteriores hasce in 
spiritibus ; unde fit, ut mea ipsa puncta materialia sint, & eorum massae constituant 
corpora a spiritibus longissime discrepantia. Si possibile sit illud substantiae genus, quod 
& hujusmodi vires activas habeat cum inertia conjunctas, & simul cogitare possit, ac velle ; 
id quidem nee corpus erit, nee spiritus, sed tertium quid, a corpore discrepans per capacitatem 
cogitationis, & voluntatis, discrepans autem a spiritu per inertiam, & vires hasce nostras, 
quae impenetrabilitatem inducunt. Sed, ut ajebam, ea quaestio hue non pertinet, & aliunde 
resolvi debet ; ut aliunde utique debet resolvi quaestio, qua quaeratur, an substantia extensa, 
& impenetrabilis [71] hasce proprietates conjungere possit cum facultate cogitandi, 

Nihil amitti, 156. Nee vero illud reponi potest, argumentum potissimum ad evincendum, materiam 

amisso argumento cogitare non posse, deduci ab extensione, & partium compositione, quibus sublatis, omne id 

eorum, qui a com- r to , . r , . VT i 

positione partium lundamentum prorsus corruere, & ad materialismum sterm viam. JNam ego sane non video, 
deducunt incapaci- quid argument! peti possit ab extensione, & partium compositione pro incapacitate cogitandi, 
& volendi. Sensibilitas, praecipua corporum, & materiae proprietas, quae ipsam adeo a 
spiritibus discriminat, non ab extensione continua, & compositione partium pendet, uti 
vidimus, sed ab impenetrabilitate, quae ipsa proprietas ab extensione continua, & compositione 
non pendet. Sunt qui adhibent hoc argumentum ad excludendam capacitatem cogitandi 
a materia, desumptum a compositione partium : si materia cogitaret ; singulae ejus partes 
deberent singulas cogitationis partes habere, adeoque nulla pars objectum perciperet ; cum 
nulla haberet earn perceptionis partem, quam habet altera. Id argumentum in mea Theoria 
amittitur ; at id ipsum, meo quidem judicio, vim nullam habet. Nam posset aliquis 
respondere, cogitationem totam indivisibilem existere in tota massa materiae, quae certa 
partium dispositione sit praedita, uti anima rationalis per tarn multos Philosophos, ut ut 
indivisibilis, in omni corpore, vel saltern in parte corporis aliqua divisibili existit, & ad 
ejusmodi praesentiam praestandam certa indiget dispositione partium ipsius corporis, qua 
semel laesa per vulnus, ipsa non potest ultra ibi esse ; atque ut viventis corporei, sive animalis 
rationalis natura, & determinatio habetur per materiam divisibilem, & certo modo 
constructam, una cum anima indivisibili ; ita ibi per indivisibilem cogitationem inhaerentem 
divisibili materise natura, & determinatio cogitantis haberetur. Unde aperte constat eo 
argumento amisso, nihil omnino amitti, quod jure dolendum sit. 

Etiam si quidpiam 157. Sed quidquid de eo argumento censeri debeat, nihil refert, nee ad infirmandam 

iam^poStive ^prob- Theoriam positivis, & validis argumentis comprobatam, ac e solidissimis principiis directa 

ari, & in ea manere ratiocinatione deductani, quidquam potest unum, vel alterum argumentum amissum, quod 

intemter1ain me & a ^ probandam aliquam veritatem aliunde notam, & a revelatis principiis aut directe, aut 

spiritum. indirecte confirmatam, ab aliquibus adhibeatur, quando etiam vim habeat aliquam, quam, 

uti ostendi, superius allatum argumentum omnino non habet. Satis est, si ilia Theoria cum 

ejusmodi veritate conjungi possit, uti haec nostra cum immaterialitate spirituum con- 

jungitur optime, cum retineat pro materia inertiam, impenetrabilitatem, sensibilitatem, 

incapacitatem cogitandi, & pro spiritibus retineat incapacitatem afHciendi sensus nostros 

per impenetrabilitatem, & facultatem cogitandi, ac volendi. [72] Ego quidem in ipsius 


of thought or will. Wherefore I also acknowledge each of those essential differences between 
matter and spirit, which are acknowledged by everyone ; but by me it is not deduced from 
extension and continuous composition, but, just as correctly, from things that can be 
conjoined with simplicity & non-extension, & can combine with them. 

155. Now if there were substances capable of thought & will that also had a law of if it were possible 
forces of this kind, is it possible that they would produce the same effects with respect to substanc<Pthatwas 
our senses, as points of this sort ? Truly, I will answer that I do not seek to know in this both endowed with 
connection, whether impenetrability & sensibility, which depend on these forces, can capsTbieofthoughT 
be conjoined with the faculty of thinking & willing ; indeed this question comes to the it would be neither 
same thing as the general idea of the relations of impenetrability of extended & composite matter nor s P' nt - 
things to the power of thinking & willing. I will say but this, that we form our ideas, 

partly from observations, of the senses in the case of bodies, & of the inner consciousness 
in the case of spirits, together with reflections upon them, partly, & indeed more especially 
in the case of spirits, from directly revealed principles, or matters closely connected with 
revealed principles ; & these ideas involve for matter impenetrability, sensibility, combined 
with incapacity for thought, & for spirit an incapacity for affecting our senses by means 
of impenetrability, together with the capacity for thinking and willing. I admit the former 
of these in the case of my points, & the latter for spirits ; so that these points of mine 
are material points, & masses of them compose bodies that are far different from spirits. 
Now if it were possible that there should be some kind of substance, which has both active 
forces of this kind together with a force of inertia & also at the same time is able to 
think and will ; then indeed it will neither be body nor spirit, but some third thing, differing 
from a body in its capacity for thought & will, & also from spirit by possessing inertia 
and these forces of mine, which lead to compenetration. But as I was saying, that question 
does not concern me now, & the answer must be found by other means. So by other 
means also must the answer be found to the question, in which we seek to know whether 
a substance that is extended & impenetrable can conjoin these two properties with the 
faculty of thinking and willing. 

156. Now it cannot be ignored that an argument of great importance in proving that Nothing is lost 
matter is incapable of thought is deduced from extension & composition by parts ; & ^ n ^g^^ 1 " 1 ^ 
if these are denied, the whole foundation breaks down, & the way is laid open to materialism, those who deduce 
But really I do not see what in the way of argument can be derived from extension & i? t&%~* y j; r 

... , *, i i i *ii* n *i "v v i_ f lutJU 5 i *^ irom com- 

composition by parts, to support incapacity for thinking and willing, bensibmty, the cruel position by parts. 

property of bodies & of matter, which is so much different from spirits, does not depend 

on continuous extension & composition by parts, as we have seen, but on impenetrability ; 

& this latter property does not depend on continuous extension & composition. There 

are some, who use the following argument, derived from composition by parts, to exclude 

from matter the capacity for thought : If matter were to think, then each of its parts 

would have a separate part of the thought, & thus no part would have perception of the 

object of thought ; for no part can have that part of the perception that another part has. 

This argument is neglected in my Theory ; but the argument itself, at least so I think, is 

unsound. For one can reply that the complete thought exists as an indivisible thing in 

the whole mass of matter, which is endowed with a certain arrangement of parts, in^the 

same way as the rational soul in the opinion of so many philosophers exists, although it is 

indivisible, in the whole of the body, or at any rate in a certain divisible part of the body ; 

& to maintain a presence of this kind there is need for a definite arrangement of the parts of 

the body, which if at any time impaired by a wound would no longer exist there. Thus, 

just as from the nature of a living body, or of a rational animal, determination arises from 

matter that is divisible & constructed on a definite plan, in conjunction with an indivisible 

mind ; so also in this case by means of indivisible thought inherent in the nature of divisible 

matter, there is a propensity for thought. From this it is very plain that, if this 

argument is dismissed, there will be nothing neglected that we have any reason to 


157. But whatever opinion we are to form about this argument, it makes no difference, Even a something 
nor can it weaken a Theory that has been corroborated by direct & valid arguments, & iheVheorrcln C be 
deduced from the soundest principles by a straightforward chain of reasoning, if we leave P^J^. in & a d t ^t 
out one or other of the arguments, which have been used by some for the purpose of ^f^fi rem ain in 
testing some truth that is otherwise known & confirmed by revealed principles either j^^fj*^ 
directly or indirectly ; even when the argument has some validity, which, as I have shown, matter & spir i t . 
that adduced above has not in any way. It is sufficient if that theory can be conjoined 

with such a truth ; just as this Theory of mine can be conjoined in an excellent manner 
with the immateriality of spirits. For it retains for matter inertia, impenetrability, 
sensibility, & incapacity for thinking, & for spirits it retains the incapacity for affecting 
our senses by impenetrability, & the faculty of thinking or willing. Indeed I assume the 


materiae, & corporeae substantias definitione ipsa assumo incapacitatem cogitandi, & volendi, 
& dico corpus massam compositam e punctis habentibus vim inertiae conjunctam cum 
viribus activis expressis in fig. i,& cum incapacitate cogitandi, ac volendi, qua definitione 
admissa, evidens est, materiam cogitare non posse ; quae erit metaphysica quaedam conclusio, 
ea definitione admissa, certissima : turn ubi solae rationes physicae adhibeantur, dicam, haec 
corpora, quae meos afficiunt sensus, esse materiam, quod & sensus afficiant per illas utique 
vires, & non cogitent. Id autem deducam inde, quod nullum cogitationis indicium 
praestent ; quae erit conclusio tantum physica, circa existentiam illius materiae ita definitae, 
aeque physice certa, ac est conclusio, quae dicat lapides non habere levitatem, quod nunquam 
earn prodiderint ascendendo sponte, sed semper e contrario sibi relict! descenderint. 

Sensus omnino fain 158. Quod autem pertmet ad ipsam corporum, & materiae ideam, quae videtur exten- 

^nultat^in^xten- si nem continuam, & contactum partium involvere, in eo videntur mihi quidem Cartesian! 
sionis, quam nobis inprimis, qui tantopere contra prasjudicia pugnare sunt visi, praejudiciis ipsis ante omnes 
alios indulsisse. Ideam corporum habemus per sensus ; sensus autem de continuitate 
accurata judicare omnino non possunt, cum minima intervalla sub sensus non cadant. Et 
quidem omnino certo deprehendimus illam continuitatem, quam in plerisque corporibus 
nobis objiciunt sensus nostri, nequaquam haberi. In metallis, in marmoribus, in vitris, 
& crystallis continuitas nostris sensibus apparet ejusmodi, ut nulla percipiamus in iis vacua 
spatiola, nullos poros, in quo tamen hallucinari sensus nostros manifesto patet, turn ex 
diversa gravitate specifica, quae a diversa multitudine vacuitatum oritur utique, turn ex 
eo, quod per ilia insinuentur substantiae plures, ut per priora oleum diffundatur, per 
posteriora liberrime lux transeat, quod quidem indicat, in posterioribus hisce potissi- 
mum ingentem pororum numerum, qui nostris sensibus delitescunt. 

Fons prajudici- 159- Quamobrem jam ejusmodi nostrorum sensuum testimonium, vel potius noster 

orum : haberi pro eor um ratiociniorum usus, in hoc ipso genere suspecta esse debent, in quo constat nos 

nulhs in se, quas , .... . 11- v -i_ 

sunt nuiia in nostris decipi. Suspican igitur licet, exactam continuitatem sine urns spatiolis, ut in majonbus 
sensibus : eorum corporibus ubique deest, licet sensus nostri illam videantur denotare, ita & in minimis 
quibusvis particulis nusquam haberi, sed esse illusionem quandam sensuum tantummodo, 
& quoddam figmentum mentis, reflexione vel non utentis, vel abutentis. Est enim 
solemne illud hominibus, atque usitatum, quod quidem est maximorum praejudiciorum 
fons, & origo praecipua, ut quidquid in nostris sensibus est nihil, habeamus pro nihilo 
absolute. Sic utique per tot saecula a multis est creditum, & nunc etiam a vulgo creditur, 
[73] quietem Telluris, & diurnum Solis, ac fixarum motum sensuum testimonio evinci, 
cum apud Philosophos jam constet, ejusmodi qusestionem longe aliunde resolvendam esse, 
quam per sensus, in quibus debent eaedem prorsus impressiones fieri, sive stemus & nos, & 
Terra, ac moveantur astra, sive moveamur communi motu & nos, & Terra, ac astra 
consistant. Motum cognoscimus per mutationem positionis, quam objecti imago habet 
in oculo, & quietem per ejusdem positionis permanentiam. Tarn mutatio, quam 
permanentia fieri possunt duplici modo : mutatio, primo si nobis immotis objectum movea- 
tur ; & permanentia, si id ipsum stet : secundo, ilia, si objecto stante moveamur nos ; haec, si 
moveamur simul motu communi. Motum nostrum non sentimus, nisi ubi nos ipsi motum 
inducimus, ut ubi caput circumagimus, vel ubi curru delati succutimur. Idcirco habemus 
turn quidem motum ipsum pro nullo, nisi aliunde admoneamur de eodem motu per causas, 
quae nobis sint cognitae, ut ubi provehimur portu, quo casu vector, qui jam diu assuevit idese 
littoris stantis, & navis promotae per remos, vel vela, corrigit apparentiam illius, terrceque 
urbesque recedunt, & sibi, non illis, motum adjudicat. 

Eorum correctio 160. Hinc Philosophus, ne fallatur, non debet primis hisce ideis acquirere, quas e 

ubi deprehendatur, se nsationibus haurimus, & ex illis deducere consectaria sine diligent! perquisitione, ac in ea 

modo al cum tl s a en n quae ab infantia deduxit, debet diligenter inquirere. Si inveniat, easdem illas sensuum 

suum apparentia perceptiones duplici modo aeque fieri posse ; peccabit utique contra Logicae etiam naturalis 

leges, si alterum modum prze altero pergat eligere, unice, quia alterum antea non viderat, 

& pro nullo habuerat, & idcirco alteri tantum assueverat. Id vero accidit in casu nostro : 


incapacity for thinking & willing in the very definition of matter itself & corporeal 
substance ; & I say that a body is a mass composed of points endowed with a force of 
inertia together with such active forces as are represented in Fig. i, & an incapacity for 
thinking & willing. If this definition is taken, it is clear that matter cannot think ; & 
this will be a sort of metaphysical conclusion, which will follow with absolute certainty 
from the acceptation of the definition. Again, where physical arguments are alone employed, 
I say that such bodies as affect our senses are matter, because they affect the senses 
by means of the forces under consideration, & do not think. I also deduce the same 
conclusion from the fact that they afford no evidence of thought. This will be a conclusion 
that is solely physical with regard to the existence of matter so defined ; & it will be just 
as physically true as the conclusion that says that stones do not possess levity, deduced from 
the fact that they never display such a thing by an act of spontaneous ascent, but on the 
contrary always descend if left to themselves. 

158. With regard to the idea of bodies & matter, which seems to involve continuous The senses are 
extension, it seems to me indeed that in this matter the Cartesians in particular, who have altogether at fault 

. r i i_ m the greatness of 

appeared to impugn pre judgments with so much vigour, have given themselves up to these the continuity of 

prejudgments more than anyone else. We obtain the idea of bodies through the senses ; f^^'^beiieve 5 '' 

and the senses cannot in any way judge on a matter of accurate continuity ; for very small 

intervals do not fall within the scope of the senses. Indeed we quite take it for granted 

that the continuity, which our senses meet with in a large number of bodies, does not really 

exist. In metals, marble, glass & crystals there appears to our senses to be continuity, 

of such sort that we do not perceive in them any little empty spaces, or pores ; but in this 

respect the senses have manifestly been deceived. This is clear, both from their different 

specific gravities, which certainly arises from the differences in the numbers of the empty 

spaces ; & also from the fact that several substances will insinuate themselves through 

their substance. For instance, oil will diffuse itself through the former, & light will pass 

quite freely through the latter ; & this indeed indicates, especially in the case of the 

latter, an immense number of pores ; & these are concealed from our senses. 

159. Hence such evidence of our senses, or rather our employment of such arguments, The origin of pre- 
must now lie open to suspicion in that class, in which it is known that we have been deceived, j^fjdered : as^o- 
We may then suspect that accurate continuity without the presence of any little empty thing, which are 
spaces such as is certainly absent from bodies of considerable size, although our senses SSe 1 srases ar^con- 
seem to remark its presence is also nowhere existent in any of their smallest particles ; cemed ; examples 
but that it is merely an illusion of the senses, & a sort of figment of the brain through its 

not using, or through misusing, reflection. For it is a customary thing for men (& a 
thing that is frequently done) to consider as absolutely nothing something that is nothing 
as far as the senses are concerned ; & this indeed is the source & principal origin of 
the greatest prejudices. Thus for many centuries it was credited by many, & still is 
believed by the unenlightened, that the Earth is at rest, & that the daily motions of the 
Sun & the fixed stars is proved by the evidence of the senses ; whilst among philosophers 
it is now universally accepted that such a question has to be answered in a far different 
manner from that by means of the senses. Exactly the same impressions are bound to be 
obtained, whether we & the Earth stand still & the stars are moved, or we & the 
Earth are moved with a common motion & the stars are at rest. We recognize motion 
by the change of position, which the image of an object has in the eye ; and rest by the 
permanence of that position. Now both the change & the permanence can come about 
in two ways. Firstly, if we remain at rest, there is a change of position if the object is 
moved, & permanence if it too is at rest ; secondly, if we move, there is a change if the 
object is at rest, & permanence if we & it move with a motion common to both. We 
do not feel ourselves moving, unless we ourselves induce the motion, as when we turn the 
head, or when we are jolted as we are borne in a vehicle. Hence we consider that the 
motion is nothing, unless we are made to notice in other ways that there is motion by causes 
that are known to us. Thus, when " we leave the harbour" a passenger who has for some time 
been accustomed to the idea of a shore remaining still, & of a ship being propelled by 
oars or sails, corrects the apparent motion of the shore ; &, as " the land & buildings recede" 
he attributes the motion to himself and not to them. 

160. Hence, the philosopher, to avoid being led astray, must not seek to obtain from ^ctionjrf ^ 
these primary ideas that we derive from the senses, or deduce from them, consequential known that the 
theorems, without careful investigation; & he must carefully study those things that matter^ ^annot^ be 
he has deduced from infancy. If he find that these very perceptions by the senses can agreement with 
come about in two ways, one of which is as probable as the other ; then he will certainly hat the is JV e 
commit an offence against the laws of natural logic, if he should proceed to choose one some other way . 
method in preference to the other, solely for the reason that previously he had not seen 

the one & took no account of it, & thus had become accustomed to the other. Now 



sensationes habebuntur eaedem, sive materia constet punctis prorsus inextensis, & distantibus 
inter se per intervalla minima, quae sensum fugiant, ac vires ad ilia intervalla pertinentes 
organorum nostrorum fibras sine ulla sensibili interruptione afficiant, sive continua sit, & 
per immediatum contactum agat. Patebit autem in tertia hujusce operis parte, quo pacto 
proprietates omnes sensibiles corporum generales, immo etiam ipsorum prsecipua discrimina, 
cum punctis hisce indivisibilibus conveniant, & quidem multo sane melius, quam in communi 
sententia de continua extensione materiae. Quamobrem errabit contra rectae ratiocinationis 
usum, qui ex praejudicio ab hujusce conciliationis, & alterius hujusce sensationum nostrarum 
causae ignoratione inducto, continuam extensionem ut proprietatem necessariam corporum 
omnino credat, & multo magis, qui censeat, materialis substantive ideam in ea ipsa continua 
extensione debere consistere. 

Ordo idearum, quas 

esse per tactum. 

161. Verum quo magis evidenter constet horum prsejudiciorum origo, afferam hie 
dissertationis De Materia Divisibilita-\j4\-te, & Principiis Corporum, numeros tres inci- 
piendo a 14, ubi sic : " utcunque demus, quod ego omnino non censeo, aliquas esse innatas 
ideas, & non per sensus acquisitas ; illud procul dubio arbitror omnino certum, ideam 
corporis, materiae, rei corporeae, rei materialis, nos hausisse ex sensibus. Porro ideas prims 
omnium, quas circa corpora acquisivimus per sensus, fuerunt omnino eae, quas in nobis 
tactus excitavit, & easdem omnium frequentissimas hausimus. Multa profecto in ipso 
materno utero se tactui perpetuo offerebant, antequam ullam fortasse saporum, aut odorum, 
aut sonorum, aut colorum ideam habere possemus per alios sensus, quarum ipsarum, ubi eas 
primum habere ccepimus, multo minor sub initium frequentia fuit. Idese autem, quas per 
tactum habuimus, ortae sunt ex phsenomenis hujusmodi. Experiebamur palpando, vel 
temere impingendo resistentiam vel a nostris, vel a maternis membris ortam, quae cum 
nullam interruptionem per aliquod sensibile intervallum sensui objiceret, obtulit nobis ideam 
impenetrabilitatis, & extensionis continuae : cumque deinde cessaret in eadem directione, 
alicubi resistentia, & secundum aliam directionem exerceretur ; terminos ejusdem quanti- 
tatis concepimus, & figurse ideam hausimus." 

Quae fuerint turn 
consideranda : in- 
fantia ad eas re- 
flexiones, inepta : in 
quo ea sita sit. 

162. " Porro oriebantur haec phsenomena a corporibus e materia jam efformatis, non a 
singulis materiae particulis, e quibus ipsa corpora componebantur. Considerandum 
diligenter erat, num extensio ejusmodi esset ipsius corporis, non spatii cujusdam, per quod 
particulae corpus efformantes diffunderentur : num ea particulse ipsae iisdem proprietatibus 
essent praeditae : num resistentia exerceretur in ipso contactu, an in minimis distantiis sub 
sensus non cadentibus vis aliqua impedimento esset, quae id ageret, & resistentia ante ipsum 
etiam contactum sentiretur : num ejusmodi proprietates essent intrinsecae ipsi materiae, ex 
qua corpora componuntur, & necessariae : an casu tantum aliquo haberentur, & ab extrinseco 
aliquo determinante. Haec, & alia sane multa considerate diligentius oportuisset : sed erat 
id quidem tempus maxime caliginosum, & obscurum, ac reflexionibus minus obviis minime 
aptum. Praster organorum debilitatem, occupabat animum rerum novitas, phaenomenorum 
paucitas, & nullus, aut certe satis tenuis usus in phaenomenis ipsis inter se comparandis, & 
ad certas classes revocandis, ex quibus in eorum leges, & causas liceret inquirere & systema 
quoddam efformare, quo de rebus extra nos positis possemus ferre judicium. Nam in hac 
ipsa phaenomenorum inopia, in hac efformandi systematis difficultate, in hoc exiguo 
reflexionum usu, magis etiam, quam in organorum imbecillitate, arbitror, sitam esse 

inde [75] 163. "In hac tanta rerum caligine ea prima sese obtulerunt animo, quae^ minus 
orta extensionis j ta jndagine, minus intentis reflexionibus indigebant, eaque ipsa ideistoties repetitis altius 

continuae ut essen- . . .- 1 

tiaiis, odorum, &c., impressa sunt, & tenacius adhaeserunt, & quendam veluti campum nacta prorsus vacuum, 
ut accidentaiium. & ac jhuc immunem, suo quodammodo jure quandam veluti possessionem inierunt. Inter- 
valla, quae sub sensum nequaquam cadebant, pro nullis habita : ea, quorum ideae^ semper 
simul conjunctae excitabantur, habita sunt pro iisdem, vel arctissimo, & necessario^ nexu 
inter se conjunctis. Hinc illud effectum est, ut ideam extensionis continuae, ideam 


that is just what happens in the case under consideration. The same sensations will be 
experienced, whether matter consists of points that are perfectly non-extended & distant 
from one another by very small intervals that escape the senses, & forces pertaining to 
those intervals affect the nerves of our organs without any sensible interruption ; or 
whether it is continuous and acts by immediate contact. Moreover it will be clearly shown, 
in the third part of this work, how all the general sensible properties of bodies, nay even 
the principal distinctions between them as well, will fit in with these indivisible points ; 
& that too, in a much better way than is the case with the common idea of continuous 
extension of matter. Wherefore he will commit an offence against the use of true reasoning, 
who, from a prejudgment derived from this agreement & from ignorance of this alter- 
native cause for our sensations, persists in believing that continuous extension is an 
absolutely necessary property of bodies ; and much more so, one who thinks that 
the very idea of material substance must depend upon this very same continuous 

161. Now in order that the source of these prejudices may be the more clearly known, Order of the ideas 
I will here quote, from the dissertation De Materice Divisibilitate & Princi-pii Corporum, ^ b^ies^tte 
three articles, commencing with Art. 14, where we have : " Even if we allow (a thing quite first ideas come 
opposed to my way of thinking) that some ideas are innate & are not acquired through o^ifch the SenSe 
the senses, there is no doubt in my mind that it is quite certain that we derive the idea 

of a body, of matter, of a corporeal thing, or a material thing, through the senses. Further, 
the very first ideas, of all those which we have acquired about bodies through the senses, 
would be in every circumstance those which have excited our sense of touch, & these 
also are the ideas that we have derived on more occasions than any other ideas. Many 
things continually present themselves to the sense of touch actually in the very womb of 
our mothers, before ever perchance we could have any idea of taste, smell, sound, or colour, 
through the other senses ; & of these latter, when first we commenced to have them, 
there were to start with far fewer occasions for experiencing them. Moreover the ideas 
which we have obtained through the sense of touch have arisen from phenomena of the 
following kind. We experienced a resistance on feeling, or on accidental contact with, an 
object ; & this resistance arose from our own limbs, or from those of our mothers. Now, 
since this resistance offered no opposition through any interval that was perceptible to the 
senses, it gave us the idea of impenetrability & continuous extension ; & then when 
it ceased in the original direction at any place & was exerted in some other direction, 
we conceived the boundaries of this quantity, & derived the idea of figure." 

162. " Furthermore, these phenomena will have arisen from bodies already formed from Such things de- 
matter, not from the single particles of matter of which the bodies themselves were composed. S^time ^tae tf 
It would have to be considered carefully whether such extension was a property of the tude of inf'ancy^for 
body itself, & not of some space through which the particles forming the body were su .f h . r eflection ; on 

j-rc JIT -11 i r i i i wnat the y ma Y be 

diffused ; whether the particles themselves were endowed with the same properties ; founded, 
whether the resistance was exerted only on actual contact, or whether, at very small 
distances such as did not fall within the scope of the senses, some force would act as a 
hindrance & produce the same effect, and resistance would be felt even before actual 
contact ; whether properties of this kind would be intrinsic in the matter of which the 
bodies are composed, & necessary to its existence ; or only possessed in certain cases, 
being due to some external influence. These, & very many other things, should have 
been investigated most carefully ; but the period was indeed veiled in mist & obscurity 
to a great degree, & very little fitted for aught but the most easy thought. In addition 
to the weakness of the organs, the mind was occupied with the novelty of things & the 
rareness of the phenomena ; & there was no, or certainly very little, use made of comparisons 
of these phenomena with one another, to reduce them to definite classes, from which it 
would be permissible to investigate their laws & causes & thus form some sort of system, 
through which we could bring the judgment to bear on matters situated outside our own 
selves. Now, in this very paucity of phenomena, in this difficulty in the matter of forming 
a system, in this slight use of the powers of reflection, to a greater extent even than in the 
lack of development of the organs, I consider that infancy consists." 

163. " In this dense haze of things, the first that impressed themselves on the mind Th ence P r J' u <^- 

i 1*1 -i i 11 i n i mcnis> di c uci i vcu. 

were those which required a less deep study & less intent investigation ; & these, since that continuity of 
the ideas were the more often renewed, made the greater impression & became fixed J^^J 1 ^ S 
the more firmly in the mind, & as it were took possession of, so to speak, a land that they continuity of odours 
found quite empty & hitherto immune, by a sort of right of discovery. Intervals, which &c - * accidental. 
in no wise came within the scope of the senses, were considered to be nothing ; those things, 
the ideas of which were always excited simultaneously & conjointly, were considered 
as identical, or bound up with one another by an extremely close & necessary bond. 
Hence the result is that we have formed the idea of continuous extension, the idea of 


impenetrabilitatis prohibentis ulteriorem motum in ipso tantum contactu corporibus 
affinxerimus, & ad omnia, quae ad corpus pertinent, ac ad materiam, ex qua ipsum constat, 
temere transtulerimus : quse ipsa cum primum insedissent animo, cum frequcntissimis, immo 
perpetuis phaenomenis, & experimentis confirmarentur ; ita tenaciter sibi invicem 
adhseserunt, ita firmiter ideae corporum immixta sunt, & cum ea copulata ; ut ea ipsa pro 
primis corporibus, & omnium corporearum rerum, nimirum etiam materiae corpora compo- 
nentis, ejusque partium proprietatibus maxime intrinsecis, & ad naturam, atque essentiam 
earundem pertinentibus, & turn habuerimus, & nunc etiam habeamus, nisi nos praejudiciis 
ejusmodi liberemus. Extensionem nimirum continuam, impenetrabilitatem ex contactu, 
compositionem ex partibus, & figuram, non solum naturae corporum, sed etiam corporeae 
materiae, & singulis ejusdem partibus, tribuimus tanquam proprietates essentiales : csetera, 
quae serius, & post aliquem reflectendi usum deprehendimus, colorem, saporem, odorem 
sonum, tanquam accidentales quasdam, & adventitias proprietates consideravimus." 

propositiones 164. Ita ego ibi, ubi Theoriam virium deinde refero, quam supra hie exposui, ac ad 

Theoriamcontinen? P r3 ecipuas corporum proprietates applico, quas ex ilia deduco, quod hie praestabo in parte 
tis. tertia. Ibi autem ea adduxeram ad probandam primam e sequentibus propositionibus, 

quibus probatis & evincitur Theoria mea, & vindicatur : sunt autem hujusmodi : i. Nullo 
prorsus argumento evincitur materiam habere extensionem continuam, W non potius constare e 
punctis prorsus indivisibilibus a se per aliquod intervallum distantibus ; nee ulla ratio seclusis 
pr&judiciis suadet extensionem ipsam continuam potius, quam compositionem e punctis prorsus 
indivisibilibus, inextensis, y nullum continuum extensum constituentibus. 2. Sunt argumenta, 
y satis valida ilia quidem, qua hanc compositionem e punctis indivisibilibus evincant extensioni 
ipsi continues pr&ferri oportere. 

Quo pacto con- 165. At quodnam extensionis genus erit istud, quod e punctis inextensis, & spatio 

coaiescan^lnmassas imaginario, sive puro nihilo [76] constat ? Quo pacto Geometria locum habere poterit, 
tenaces: transitus ubi nihil habetur reale continue extensum? An non punctorum ejusmodi in vacuo 
dam Partem secun " innatantium congeries erit, ut quaedam nebula unico oris flatu dissolubilis prorsus sine ulla 
consistent! figura, solidate, resistentia ? Haec quidem pertinent ad illud extensionis ,& 
cohaesionis genus, de quo agam in tertia parte, in qua Theoriam applicabo ad Physicam, ubi 
istis ipsis difficultatibus faciam satis. Interea hie illud tantummodo innuo in antecessum, me 
cohaesionem desumere a limitibus illis, in quibus curva virium ita secat axem, ut a repulsione 
in minoribus distantiis transitus fiat ad attractionem in majoribus. Si enim duo puncta 
sint in distantia alicujus limitis ejus generis, & vires, quae immutatis distantiis oriuntur, sint 
satis magnae, curva secante axem ad angulum fere rectum, & longissime abeunte ab ipso ; 
ejusmodi distantiam ea puncta tuebuntur vi maxima ita, ut etiam insensibiliter compressa 
resistant ulteriori compressioni, ac distracta resistant ulteriori distractioni ; quo pacto si 
multa etiam puncta cohaereant inter se, tuebuntur utique positionem suam, & massam 
constituent formae tenacissimam, ac eadem prorsus phsenomena exhibentem, quae exhiberent 
solidae massulae in communi sententia. Sed de hac re uberius, uti monui, in parte tertia : 
nunc autem ad secundam faciendus est gradus. 


impenetrability preventing further motion only on the absolute contact of bodies ; & 
then we have heedlessly transferred these ideas to all things that pertain to a solid body, 
and to the matter from which it is formed. Further, these ideas, from the time when they 
first entered the mind, would be confirmed by very frequent, not to say continual, phenomena 
& experiences. So firmly are they mutually bound up with one another, so closely are 
they intermingled with the idea of solid bodies & coupled with it, that we at the time 
considered these two things as being just the same as primary bodies, & as peculiarly 
intrinsic properties of all corporeal things, nay further, of the very matter from which 
bodies are composed, & of its parts ; indeed we shall still thus consider them, unless we 
free ourselves from prejudgments of this nature. To sum up, we have attributed continuous 
extension, impenetrability due to actual contact, composition by parts, & shape, as if 
they were essential properties, not only to the nature of bodies, but also to corporeal matter 
& every separate part of it ; whilst others, which we comprehend more deeply & as 
a consequence of some considerable use of thought, such as colour, taste, smell & sound, 
we have considered as accidental or adventitious properties." 

164. Such are the words I used ; & then I stated the Theory of forces which I have A pair of proposi- 
expounded in the previous articles of this work, and I applied the theory to the principal tation 0f containing 
properties of bodies, deducing them from it ; & this I will set forth in the third part the whole of y 
of the present work. In the dissertation I had brought forward the arguments quoted *" 

in order to demonstrate the truth of the first of the following theorems. If these theorems 
are established, then my Theory is proved & verified; they are as follows : i. There is 
absolutely no argument that can be brought forward to prove that matter has continuous extension, 
y that it is not rather made up of perfectly indivisible points separated from one another by 
a definite interval ; nor is there any reason apart from prejudice in favour of continuous extension 
in preference to composition from points that are perfectly indivisible, non-extended, & forming 
no extended continuum of any sort. 2. There are arguments, W fairly strong ones too, which 
will prove that this composition from indivisible points is preferable to continuous extension. 

165. Now what kind of extension can that be which is formed out of non-extended The manner in 

o t i 5 TT /-i 1111 which groups of 

points & imaginary space, i.e., out of pure nothing ? How can Geometry be upheld points coalesce into 
if no thing is considered to be actually continuously extended ? Will not groups of points, tenacious masses : 

n t i 11-1 i i T i i i i n & then we pass on 

floating in an empty space of this sort be like a cloud, dissolving at a single breath, & to the second part. 

absolutely without a consistent figure, or solidity, or resistance ? These matters pertain 

to that kind of extension & cohesion, which I will discuss in the third part, where I apply 

my Theory to physics & deal fully with these very difficulties. Meanwhile I will here 

merely remark in anticipation that I derive cohesion from those limit-points, in which the 

curve of forces cuts the axis, in such a way that a transition is made from repulsion at smaller 

distances to attraction at greater distances. For if two points are at the distance that 

corresponds to that of any of the limit-points of this kind, & the forces that arise when 

the distances are changed are great enough (the curve cutting the axis almost at right angles 

& passing to a considerable distance from it), then the points will maintain this distance 

apart with a very great force ; so that when they are insensibly compressed they will resist 

further compression, & when pulled apart they resist further separation. In this way 

also, if a large number of points cohere together, they will in every case maintain their 

several positions, & thus form a mass that is most tenacious as regards its form ; & this 

mass will exhibit exactly the same phenomena as little solid masses, as commonly understood, 

exhibit. But I will discuss this more fully, as I have remarked, in the third part ; for now 

we must pass on to the second part. 

[77] PARS II 
Theories *Applicato ad Mechanicam 

Ante appHcatipnem 166. Considerabo in hac secunda parte potissimum generates quasdam leges aequilibrii 
consideratio'curvs! & motus tam punctorum, quam massarum, quae ad Mechanicam utique pertinent, & ad 
plurima ex iis, quae in elementis Mechanics passim traduntur, ex unico principio, & adhibito 
constant! ubique agendi modo, demonstranda viam sternunt pronissimam. Sed prius 
praemittam nonnulla quae pertinent ad ipsam virium curvam, a qua utique motuum, 
phaenomena pendent omnia. 

Quid in ea con- 167. In ea curva consideranda sunt potissimum tria, arcus curvae, area comprehensa 
siderandum. i nter axemj & arcum, quam general ordinata continue fluxu, ac puncta ilia, in quibus 

curva secat axem. 

Diversa arcnum 1 68. Quod ad arcus pcrtinet, alii dici possunt repulsivi, & alii attractivi, prout nimirum 

asymptotic! "tiam J acent ac * partes cruris asymptotici ED, vel ad contrarias, ac terminant ordinatas exhibentes 

numero infiniti. vires repulsivas, vel attractivas. Primus arcus ED debet omnino esse asymptoticus ex 

parte repulsiva, & in infinitum productus : ultimus TV, si gravitas cum lege virium 

reciproca duplicata distantiarum protenditur in infinitum, debet itidem esse asymptoticus 

ex parte attractiva, & itidem natura sua in infinitum productus. Reliquos figura I exprimit 

omnes finitos. Verum curva Geometrica etiam ejus naturae, quam exposuimus, posset habere 

alia itidem asymptotica crura, quot libuerit, ut si ordinata mn in H abeat in infinitum. 

Sunt nimirum curvae continuae, & uniformis naturae, quae asymptotes habent plurimas, 

& habere possunt etiam numero infinitas. (') 

Arcus intermedii. [78] 169. Arcus intermedii, qui se contorquent circa axem, possunt etiam alicubi, 
ubi ad ipsum devenerint, retro redire, tangendo ipsum, atque id ex utralibet parte, & 
possent itidem ante ipsum contactum inflecti, & redire retro, mutando accessum in recessum, 
ut in fig. i. videre est in arcu P^R. 

Arcus prostremus 170. Si gravitas gencralis legem vis proportionalis inverse quadrate distantiae, quam 
36 non accurate servat, sed quamproxime, uti diximus in priore parte, retinet ad sensum non 
mutatam solum per totum planetarium, & cometarium systema, fieri utique poterit, ut 
curva virium non habeat illud postremum crus asymptoticum TV, habens pro asymptoto 
ipsam rectam AC, sed iterum secet axem, & se contorqueat circa ipsum.(*) Turn vero inter 

(i) S* ex. gr. in fig. 12. cyclois continua CDEFGH (3e., quam generet punctum peripheries circuli continue revoluti 
supra rectam AB, qute natura sua protenditur utrinque in infinitum, adeoque in infinitis punctis C, E, G, I, &c. occurrit 
basi AB. Si ubicunque ducatur qutevis ordinata PQ, productaturque in R ita, ut sit PR tertia post PQ, y datam quampiam 
rectam ; punctum R frit ad curvam continuum constantem totidem ramis MNO, VXY, yr., quot erunt arcus Cycloidales 
CDE, EFG, i3c,, quorum ramorum singuli habebunt bina crura asymptotica, cum ordinata PQ in accessu ad omnia puncta, 
C, E, G, &c. decrescat ultra quoscunque Unites, adeoque ordinata PR crescat ultra limites quoscunque. Erunt hie quidem 
omnes asymptoti CK, EL, GS &c. parallels inter se, & perpendiculares basi AB, quod in aliis curvis non est necessarium, 
cum etiam divergentes utcunque possint esse. Erunt autem y totidem numero, quot puncta. ilia C, E, G &c., nimirum 
infinite. Eodem autem pacto curvarum quarumlibet singuli occursus cum axe in curvis per eas hac eadem lege genitis 
bina crura asymptotica generant, cruribus ipsis jacentibus, vel, ut hie, ad eandem axis partem, ubi curva genetrix ab eo 
regreditur retro post appulsum, vel etiam ad partes oppositas, ubi curva genetrix ipsum secet, ac transiliat : cumque possit 
eadem curva altiorum generum secari in punctis plurimis a recta, vel contingi ; poterunt utique haberi y rami asymptotici 
in curva eadem continua, quo libuerit data numero. 

(k)Nam ex ipsa Geometrica continuitate, quam persecutus sum in dissertatione De Lege Continuitatis, y in dissertatione 
De Transformatione Locorum Geometricorum adjecta Sectionibus Conicis, exhibui necessitatem generalem secundi 
illius cruris asymptotici redeuntis ex infinite. Quotiescunque enim curva aliqua saltern algebraica habet asymptoticum 
crus aliquod, debet necessario habere y alterum ipsi respondens, y habens pro asymptoto eandem rectam : sed id habere 












^Application of the Theory to Mechanics 

1 66. I will consider in this second part more especially certain general laws of Consideration of 
equilibrium, & motions both of points & masses ; these certainly belong to the science of proceeding w^t'h 
Mechanics, & they smooth the path that is most favourable for proving very many of those tne application to 
theorems, that are everywhere expounded in the elements of Mechanics, from a single 

principle, & in every case by the constant employment of a single method of dealing with 
them. But, before I do that, I will call attention to a few points that pertain to the curve 
of forces itself, upon which indeed all the phenomena of motions depend. 

167. With regard to the curve, there are three points that are especially to be considered ; The points we have 
namely, the arcs of the curve, the area included between the axis & the curve swept out regard'tolt. 1 

by the ordinate by its continuous motion, & those points in which the curve cuts the axis. 

1 68. As regards the arcs, some may be called repulsive, & others attractive, according The different kinds 
indeed as they lie on the same side of the axis as the asymptotic branch ED or on the opposite totkfarc's may even 
side, & terminate ordinates that represent repulsive or attractive forces. The first arc be infinite in num- 
ED must certainly be asymptotic on the repulsive side of the axis, & continued indefinitely. r ' 

The last arc TV, if gravity extends to indefinite distances according to a law of forces in 
the inverse ratio of the squares of the distances, must also be asymptotic on the attractive 
side of the axis, & by its nature also continued indefinitely. All the remaining arcs are 
represented in Fig. I as finite. But a geometrical curve, of the kind that we have expounded, 
may also have other asymptotic branches, as many in number as one can wish ; for instance, 
suppose the ordinate mn at H to go away to infinity. There are indeed curves, that are 
continuous & uniform, which have very many asymptotes, & such curves may even 
have an infinite number of asymptotes. (') 

169. The intermediate arcs, which wind about the axis, can also, at any point where intermediate arcs, 
they reach it, return backwards & touch it ; and they can do this on either side of it ; they 

may also be reflected and recede before actual contact, the approach being altered into a 
recession, as is to be seen in Fig. i with regard to the arc P^/yR. 

170. If universal gravity obeys the law of a force inversely proportional to the square of The ultimate arc 

the distance (which, as I remarked in the first part, it only obeys as nearly as possible, but [ t P y eS poss 

not exactly), sensibly unchanged only throughout the planetary & cometary system, it will asymptotic, 
certainly be the case that the curve of forces will not have the last arm PV asymptotic with 
the straight line AC as the asymptote, but will again cut the axis & wind about it. (*) Then 

(i) Let, for example, in Fig. 12, CDEFGH &c. be a. continuous cycloid, generated by a point on the circumference 
of a circle rolling continuously along the straight line AB ; this by its nature extends on either side to infinity, W thus 
meets the base AB in an infinite number of points such as C, E, G, I, &c. // at every point there is drawn an ordinate 
such as PQ, and this is produced to R, so that PR is a third proportional to PQ W some given straight line ; then the point 
R will trace out a continuous curve consisting of as many branches, MNO, VXY, &c., as there are cycloidal arcs, CDE, 
EFG, &c. ; each of these branches will have a pair of asymptotic arms, since the ordinate PQ on approaching any 
one of the points C,E,G, &c., will decrease beyond all limits, (3 thus the ordinate PR will increase beyond all limits. 
In this curve then there will be CK, EL, GS, &c., all asymptotes parallel to one another & perpendicular to the base 
AB ; this is not necessarily the case in other curves, since they may be also inclined to one another in any manner. 
Further they will be as many in number as there are points such as C, E, G, &c., that is to say, infinite. Again, in 
a similar way, the several intersections of any curves you please with the axis give rise to a pair of asymptotic arms 
in curves derived from them according to the same law ; and these arms lie, either on the same side of the axis, as 
in this case, where the original curve leaves the axis once more after approaching it, or indeed on opposite sides of the 
axis, where the original curve cuts W crosses it. Also, since it is possible for the same curve of higher orders to be 
cut in a large number of points, or to be touched, there will possibly be also asymptotic arms in this same continuous 
curve equal to any given number you please. 

(k) For, from the principle of geometrical continuity itself, which I discussed in my dissertation De Lege Continuitatis 
and in the dissertation De Transformatione Locorum Geometricorum appended to my Sectionum Conicarum 
Elementa, / showed the necessity for the second asymptotic arm returning from infinity. For as often as an algebraical 
curve has at least one asymptotic arm, it must also have another that corresponds to it y has the same straight line 



alios casus innumeros, qui haberi possent, unum censeo speciminis gratia hie non omitten- 
dum ; incredibile enim est, quam ferax casuum, quorum singuli sunt notatu dignissimi, 
unica etiam hujusmodi curva esse possit. 

shnufum curTserte I 7 I> Si in % H * n axe C'C sint segmenta AA', A'A" numero quocunque, quorum 

Mundoru'm mag- posteriora sint in immensum majora respectu praecedentium, & per singula transeant, 
donaikfm propor " asympto-[79]-ti AB, A'B', A"B" perpendiculares axi ; possent inter binas quasque asymptotes 
esse curvae ejus formae, quam in fig. I habuimus, & quae exhibetur hie in DEFI &c., D'E'F'F, 
&c., in quibus primum crus ED esset asymptoticum repulsivum, postremum SV attractivum, 
in singulis vero intervallum EN, quo arcus curvae contorquetur, sit perquam exiguum 
respectu intervalli circa S, ubi arcus diutissime perstet proximus hyperbolae habenti 
ordinatas in ratione reciproca duplicata distantiarum, turn vero vel immediate abiret 
in arcum asymptoticum attractivum, vel iterum contorqueretur utcunque usque ad 
ejusmodi asymptoticum attractivum arcum, habente utroque asymptotico arcu aream 
infinitam ; in eo casu collocate quocunque punctorum numero inter binas quascunque 
asymptotes, vel inter binaria quotlibet, & rite ordinato, posset exurgere quivis, ut ita 
dicam, Mundorum numerus, quorum singuli essent inter se simillimi, vel dissimillimi, 
prout arcus EF&cN, E'F'&cN' essent inter se similes, vel dissimiles, atque id ita, ut quivis 
ex iis nullum haberet commercium cum quovis alio ; cum nimirum nullum punctum 
posset egredi ex spatio incluso iis binis arcubus, hinc repulsive, & inde attractive ; & ut 
omnes Mundi minorum dimensionum simul sumpti vices agerent unius puncti respectu 
proxime majoris, qui constaret ex ejusmodi massulis respectu sui tanquam punctualibus, 
dimensione nimirum omni singulorum, respectu ipsius, & respectu distantiarum, ad quas 
in illo devenire possint, fere nulla ; unde & illud consequi posset, ut quivis ex ejusmodi 
tanquam Mundis nihil ad sensum perturbaretur a motibus, & viribus Mundi illius majoris, 
sed dato quovis utcunque magno tempore totus Mundus inferior vires sentiret a quovis 
puncto materiae extra ipsum posito accedentes, quantum libuerit, ad aequales, & parallelas 
quae idcirco nihil turbarent respectivum ipsius statum internum. 

Omissis subiimiori- 172. Sed ea jam pertinent ad applicationem ad Physicam, quae quidem hie innui 

areas pr0greSSUS ad tantumm do, ut pateret, quam multa notatu dignissima considerari ibi possent, & quanta 
sit hujusce campi fcecunditas, in quo combinationes possibiles, & possibiles formae sunt 
sane infinities infinitae, quarum, quae ab humana mente perspici utcunque possunt, ita 
sunt paucae respectu totius, ut haberi possint pro mero nihilo, quas tamen omnes unico 
intuitu prsesentes vidit, qui Mundum condidit, DEUS. Nos in iis, quae consequentur, 
simpliciora tantummodo qusedam plerumque consectabimur, quae nos ducant ad phaeno- 
mena iis conformia, quae in Natura nobis pervia intuemur, & interea progrediemur ad 
areas arcubus respondentes. 

Cuicunque axis 173. Aream curvae propositae cuicunque, utcunque exiguo, axis segmento respondentem 

aream e "respondere P osse ess e utcunque magnam, & aream respondentem cuicunque, utcunque magno, [80] 

utcunque magnam posse esse utcunque parvam, facile patet. Sit in fig. 15, MQ segmentum axis utcunque 

secundjT" 1 de^non- parvum, vel magnum ; ac detur area utcunque magna, vel parva. Ea applicata ad MQ 

stratio. exhibebit quandam altitudinem MN ita, ut, ducta NR parallela MQ, sit MNRQ aequalis 

areae datae, adeoque assumpta QS dupla QR, area trianguli MSQ erit itidem aequalis areae 

datae. Jam vero pro secundo casu satis patet, posse curvam transire infra rectam NR, 

uti transit XZ, cujus area idcirco esset minor, quam area MNRQ ; nam esset ejus pars. 

potest vel ex eadem parte, vel ex opposita ; W crus ipsum jacere potest vel ad easdem plagas partis utriuslibet cum priore 
crure, vel ad oppositas, adeoque cruris redeuntis ex infinite poshiones quatuor esse possunt. Si in fig. 13 crus ED abeat 
in infinitum, existente asymptoto ACA', potest regredi ex parte A vel ut HI, quod crus facet ad eandem plagam, velut 
KL, quod, facet ad oppositam ; y ex parte A', vel ut MN, ex eadem plaga, vel ut OP, ex opposita. In posteriore ex 
iis duabus dissertationibus profero exempla omnium ejusmodi regressuum ; ac secundi, ($ quarti casus exempla exhibet 
etiam superior genesis, si curva generans contingat axem, vel secet, ulterius progressa respectu ipsius. Inde autem fit, ut 
crura asymptotica rectilineam babentia asymptotum esse non possint, nisi numero part, ut & radices imaginarite in 
eequationibus algebraicis. 

Verum hie in curva virium, in qua arcus semper debet progredi, ut singulis distantiis, sive abscissis, singula vires, 
sive ordinatts respondeant, casus primus, & tertius haberi non possunt. Nam ordinata RQ cruris DE occurreret alicubi 
in S, S' cruribus etiam HI, MN , adeoque relinquentur soli quartus, & secundus, quorum usus erit infra. 








O \ (0 



there is one, out of an innumerable number of other cases that may possibly happen, which 
I think for the sake of an example should not be omitted here ; for it is incredible how 
prolific in cases, each of which is well worth mentioning, a single curve of this kind can be. 

171. If, in Fig. 14, there are any number of segments AA', A'A", of which each that A series of similar 
follows is immensely great with regard to the one that precedes it ; & if through each c " rve . s - wlth a s ^ ies 
point there passes an asymptote, such as AB, A'B', A"B", perpendicular to the axis ; then tionai in magnitude, 
between any two of these asymptotes there may be curves of the form given in Fig. i. 
These are represented in Fig. 14 by DEFI &c., D'E'F'I' &c. ; & in these the first arm E 
would be asymptotic & repulsive, & the last SV attractive. In each the interval EN, 
where the arc of the curve is winding, is exceedingly small compared with the interval 
near S, where the arc for a very long time continues closely approximating to the form 
of the hyperbola having its ordinates in the inverse ratio of the squares of the distances ; 
& then, either goes off straightway into an asymptotic & attractive arm, or once more 
winds about the axis until it becomes an asymptotic attractive arc of this kind, the area 
corresponding to either asymptotic arc being infinite. In such a case, if a number of points 
are assembled between any pair of asymptotes, or between any number of pairs you please, 
correctly arranged, there can, so to speak, arise from them any number of universes, 
each of them being similar to the other, or dissimilar, according as the arcs EF . . . . N, 
E'F' .... N' are similar to one another, or dissimilar ; & this too in such a way that 
no one of them has any communication with any other, since indeed no point can possibly 
move out of the space included between these two arcs, one repulsive & the other 
attractive ; & such that all the universes of smaller dimensions taken together would 
act merely as a single point compared with the next greater universe, which would 
consist of little point-masses, so to speak, of the same kind compared with itself, that is 
to say, every dimension of each of them, compared with that universe & with respect to 
the distances to which each can attain within it, would be practically nothing. From 
this it would also follow that any one of these universes would not be appreciably influenced 
in any way by the motions & forces of that greater universe ; but in any given time, 
however great, the whole inferior universe would experience forces, from any point of matter 
placed without itself, that approach as near as possible to equal & parallel forces ; these 
therefore would have no influence on its relative internal state. 

172. Now these matters really belong to the application of the Theory to physics ; & Leaving out more 
indeed I only mentioned them here to show how many things there may be well worth abstruse matters, 

j. . -i a i .,..,. ' - 9 . n i i r we pass on to areas. 

considering in that section, & how great is the fertility of this field of investigation, m 
which possible combinations & possible forms are truly infinitely infinite ; of these, those 
that can be in any way comprehended by the human intelligence are so few compared 
with the whole, that they can be considered as a mere nothing. Yet all of them were seen 
in clear view at one gaze by GOD, the Founder of the World. We, in what follows, will 
for the most part investigate only certain of the more simple matters which will lead us 
to phenomena in conformity with those things that we contemplate in Nature as far as 
our intelligence will carry us ; meanwhile we will proceed to the areas corresponding to 
the arcs. 

173. It is easily shown that the area corresponding to any segment of the axis, however To any segment of 
small, can be anything, no matter how great ; & the area corresponding to any segment, corrt'spo^a'rfy 
however great, can be anything, no matter how small. In Fig. 1 5 , let MQ be a segment of the area, however 
axis, no matter how small, or great; & let an area be given, no matter how great, or SSi ; proof^the 
small. If this area is applied to MQ a certain altitude MN will be given, such that, if NR second part of this 
is drawn parallel to MQ, then MNRQ will be equal to the given area ; & thus, if QS is a 

taken equal to twice QR, the area of the triangle MSQ will also be equal to the given area. 
Now, for the second case it is sufficiently evident that a curve can be drawn below the 
straight line NR, in the way XZ is shown, the area under which is less than the area MNRQ ; 

as its asymptote ; & this can take place with either the same part of the line or with the other part ; also the arm 
itself can lie either on the same side of either of the two parts, or on the opposite side. Thus there may be four positions 
of the arm that returns from infinity. If, in Fig. 13, the arm ED goes off to infinity, the asymptote being ACA, 
it may return from the direction of A, either like HI, wheie the arm lies on the same side of the asymptote or as KL 
which lies 'on the opposite side of it ; or from the direction of A', either as MN, on the same side, or as, DP, on the 
opposite side. In the second of these two dissertations, I have given examples of all regressions of this sort ; y the 
method of generation given above will yield examples of the second W fourth cases, if the generating curve touches 
the axis, or cuts it & passes over beyond it. Further, it thus comes about that asymptotic arms having a rectilinear 
asymptote cannot exist except in pairs, just like imaginary roots in algebraical equations. 

But here in the curve of forces, in which the arc must always proceed in such a manner that to each distance or 
abscissa there corresponds a single force or ordinate, the first tf third cases cannot occur. For the ordinate RQ of the 
arm DE would meet somewhere, in S, S', the branches HI, MN as well. Hence only the fourth & second cases are 
left ; W these we will make use of later. 



Demonst ratio 

Aream asympto- 
ticam posse esse 
infinitam, vel fini- 
tam magnitudinis 

Areas exprimere 
incrementa, vel 
decrementa quad- 
ati velocitatis. 

Quin immo licet ordinata QV sit utcunque magna ; facile patet, posse arcum MaV ita 
accedere ad rectas MQ, QV ; ut area inclusa iis rectis, & ipsa curva, minuatur infra 
quoscunque determinatos limites. Potest enim jacere totus arcus intra duo triangula 
QaM, QaV, quorum altitudines cum minui possint, 
quantum libuerit, stantibus basibus MQ, QV, potest 
utique area ultra quoscunque limites imminui. Pos- 
set autem ea area esse minor quacunque data ; 
etiamsi QV esset asymptotus, qua de re paullo 

174. Pro primo autem casu vel curva secet axem 
extra MQ, ut in T, vel in altero extremo, ut in M ; 
fieri poterit, ut ejus arcus TV, vel MV transeat per 
aliquod punctum V jacens ultra S, vel etiam per 
ipsum S ita, ut curvatura ilium ferat, quemad- 
modum figura exhibet, extra triangulum MSQ, quo 
casu patet, aream curvae respondentem intervallo MQ 
fore majorem, quam sit area trianguli MSQ, adeoque 
quam sit area data ; erit enim ejus trianguli area 
pars areae pertinentis ad curvam. Quod si curva 
etiam secaret alicubi axem, ut in H inter M, & Q, 
turn vero fieri posset, ut area respondens alteri e 
segmentis MH, QH esset major, quam area data . 

simul, & area alia assumpta, qua area assumpta esset minor area respondens segmento, 
alteri adeoque excessus prioris supra posteriorem remaneret major, quam area data. 

175. Area asymptotica clausa inter asymptotum, & ordinatam quamvis, ut in fig. I 
BA#g, potest esse vel infinita, vel finita magnitudinis cujusvis ingentis, vel exiguae. Id 
quidem etiam geometrice demonstrari potest, sed multo facilius demonstratur calculo 
integrali admodum elementari ; & in Geometriae sublimioris elementis habentur theoremata, 
ex quibus id admodum facile deducitur 0. Generaliter nimi-[8l]-rum area ejusmodi 
est infinita ; si ordinata crescit in ratione reciproca abscissarum simplici, aut majore : & 
est finita ; si crescit in ratione multiplicata minus, quam per unitatem. 

176. Hoc, quod de areis dictum est, necessarium fuit ad applicationem ad Mechanicam, 
ut nimirum habeatur scala quaedam velocitatum, quae in accessu puncti cujusvis ad aliud 
punctum, vel recessu generantur, vel eliduntur ; prout ejus motus conspiret directione vis, 
vel sit ipsi contrarius. Nam, quod innuimus & supra in adnot. (/) ad num. 118., ubi vires 
exprimuntur per ordinatas, & spatia per abscissas, area, quam texit ordinata, exprimit 
incrementum, vel decrementum quadrati velocitatis, quod itidem ope Geometrise demon- 
stratur facile, & demonstravi tam in dissertatione De Firibus Vivis, quam in Stayanis 
Supplements ; sed multo facilius res conficitur ope calculi integralis. () 

M H 

FIG. 15. 

(1) Sit Aa in Fig. I =x, ag=y ; ac sit #"y = I ; erit y = *-" >/ ", y dx elementum areee=x~ m/ *dx, cujus integrate 

*fn + A, addita constanti A, sive ob x~* > " > =y, habebitur ?xy + A. Quoniam incipit area in A, in 
n~m " n-m 

origine abscissarum ; si nm fuerit numerus positivus, adeoque n major, quam m ; area erit finita, ac valor A =o; 
area vero erit ad rectangulum AaXag, ut in ad n m, quod rectangulum, cum ag possit esse magna, & parva, ut libuerit, 
potest esse magnitudinis cujusvis. Is valor fit infinitus, si facto m =n, divisor evaaatQ; adeoque multo magis fit 
infinitus valor area, si m sit major, quam n. Unde constat, aream fore infiniiam, quotiescunque ordinatte crescent in 
ratione reciproca simplici, y majore ; secus fore finitam. 

(m) Sit u vis, c celeritas, t tempus, s spatium : erit u at = dc, cum celeritatis incrementum sit proportionale vi, W 
tempusculo ; ac erit c dt = ds, cum spatiolum confectwm respondeat velocitati, & tempusculo. Hinc eruitur dt = , 

W pariter dt =, adeoque- = W c dc = u ds. Porro 2c dc est incrementum quadrati vekcitatis cc, i3 u ds 

c u c 

in bypotbesi, quod ordinata sit w, & spatium s sit abscissa, est areola respondens spatiolo ds confecto. Igitur incrementum 
quadrati velocitatis conspirante vi, adeoque decrementum vi contraria, respondet arete respondent spatiolo percurso quovis 
infinitesimo tempusculo ; & proinde tempore etiam quovis finito incrementum, vel decrementum quadrati velocitatis 
respondet arece pertinenti ad partem axis referentem spatium percursum. 

Hinc autem illud sponte consequitur : si per aliquod spatium vires in singulis punctis eeedem permaneant, mobile autem 
adveniat cum velocitate quavis ad ejus initium ; diferentiam quadrati velocitatis finalis a quadrate velocitatis initialis 
fore semper eandem, quts idcirco erit tola velocitas finalis in casu, in quo mobile initio illius spatii haberet velocitatem 
nullam. Quare, quod nobis erit inferius usui, quadratum velocitatis finalis, conspirante vi cum directione motus, tzquabitur 
binis quadratis binarum velocitatum, ejus, quam babuit initio, W ejus,.quam acquisivisset in fine, si initio ingressum fuisset 
sine ulla velocitate. 



for it is part of it. Again, although the ordinate QV may be of any size, however great, 
it is easily shown that an arc MoV can approach so closely to the straight lines MQ, 
QV that the area included between these lines & the curve shall be diminished beyond 
any limits whatever. For it is possible for the curve to lie within the two triangles QaM, 
QaV ; & since the altitudes of these can be diminished as much as you please, whilst the 
bases MQ, QV remain the same, therefore the area can indeed be diminished beyond all 
limits whatever. Moreover it is possible for this area to be less than any given area, even 
although QV should be an asymptote ; we will consider this a little further on. 

174. Again, for the first case, either the curve will cut the axis beyond MQ, as at T, 
or at either end, as at M. Then it is possible for it to happen that an arc of it, TV or MV, 
will pass through some point V lying beyond S, or even through S itself, in such a way 
that its curvature will carry it, as shown in the diagram, outside the triangle MSQ ; in 
this case it is clear that the area of the curve corresponding to the interval MQ will be 
greater than the area of the triangle MSQ, & therefore greater than the given area, 
for the area of this triangle is part of the area belonging to the curve. But if the curve 
should even cut the axis anywhere, as at H, between M & Q, then it would be possible 
for it to come about that the area corresponding to one of the two segments MH, QH would 
be greater than the given area together with some other assumed area ; & that the area 
corresponding to the other segment should be less than this assumed area ; and thus the 
excess of the former over the latter would remain greater than the given area. 

175. An asymptotic area, bounded by an asymptote & any ordinate, like BAag in 
Fig. i, can be either infinite, or finite of any magnitude either very great or very small. 
This can indeed be also proved geometrically, but it can be demonstrated much more 
easily by an application of the integral calculus that is quite elementary ; & in the elements 
of higher geometry theorems are obtained from which it is derived quite easily. In 
general, it is true, an area of this kind is infinite ; namely when the ordinate increases in 
the simple inverse ratio of the abscissse, or in a greater ratio ; and it is finite, if it increases 
in this ratio multiplied by something less than unity. 

176. What has been said with regard to areas was a necessary preliminary to the 
application of the Theory to Mechanics ; that is to say, in order that we might obtain a 
diagrammatic representation of the velocities, which, on the approach of any point to 
another point, or on recession from it, are produced or destroyed, according as its motion 
is in the same direction as the direction of the force, or in the opposite direction. For, 
as we also remarked above, in note (/) to Art. 118, when the forces are represented by 
ordinates & the distances by abscissae, the area that the ordinate sweeps out represents 
the increment or decrement of the square of the velocity. This can also be easily proved 
by the help of geometry ; & I gave the proof both in the dissertation De Firibus Fivis 
& in the Supplements to Stay's Philosophy ; but the matter is much more easily made 
out by the aid of the integral calculus. () 


of the first 

An a s y m p totic 
area may be either 
infinite or equal to 
any finite area 

The areas represent 
the increments or 
decrements of the 
square of the velo- 

(1) In Fig. iletAa = x,ag = y; y let x m y" = I. Then will y x~ 

the element of area y dx = x~ m/ * 

dx : the integral of this is - x <-"/"+ A, where a constant A is added ; or, since x~ m/ *=y, we shall have-^ X v + A 

n-m n-m ' 

Now, since the area is initially A, at the origin of the abscissa, if n-m happened to be a positive number, y 
thus n greater than m, then the area will be finite, y the value of A will be = o. Also the area will be to 
the rectangle as n is to n-m ; y this rectangle, since ag can be either great or small, as you please, may be 
of any magnitude whatever. The value is infinite, if by making m equal to n the divisor becomes equal to zero ; & 
thus the value of the area becomes all the more infinite, if m is greater than n. Hence it follows that the area will 
be infinite, whenever the ordinates increase in a simple inverse ratio, or in a greater ratio ; otherwise it will be finite. 

(m) Let u be the force, c the velocity, t the time, y s the distance. Then will u dt dc, since the increment 
of the velocity is proportional to the force, y to the small interval of time. Also c dt = ds, since the distance traversed 
corresponds with the velocity W the small interval of time. Hence it follows that dt = dc/u, y similarly dt ds/c, 
y therefore dc/u = ds/c, & c dc u ds. Further, ^c dc is the increment of the square of the velocity c', y u ds, 
on the hypothesis that the ordinate represents u, y the abscissa the distance s, is the small area corresponding to the 
small distance traversed. Hence the increment of the square of the velocity, when in the direction of the force, y 
the decrement when opposite in direction to the force, is represented by the area corresponding to ds, the small distance 
traversed in any infinitely short time. Hence also, in any finite interval of time, the increment or decrement of the 
square of the velocity will be represented by the area corresponding to that part of the axis which represents the distance 

Hence also it follows immediately that, if through any distance the force on each of the points remains as before, 
but the moving body arrives at the beginning of it with any velocity, then the difference between the square of the final 
velocity y the square of the initial velocity will always be the same ; y this therefore will be the total final velocity, 
in the case where the moving body had no velocity at the beginning of the distance. Hence, the square of the final 
velocity, when the motion is in the same direction as the force, will be equal to the sum of the squares of the velocity which 
it had at the beginning y of the velocity it would have acquired at the end, if it had at the beginning started without 
any velocity ; a theorem that we shall make use of later. 

I 4 2 


Atque id ips u m, 
licet segmenta axis 
sint dimidia spatio- 
rum percursorum a 
singulis punctis. 

Si arese sint partim 
attractivae, partim 
repulsivae, assumen- 
dam esse differen- 
tiam earundem. 

177. Duo tamen hie tantummodo notanda sunt ; primo quidem illud : si duo 
puncta ad se invicem accedant, vel a se invicem recedant in ea recta, quae ipsa conjungit, 
segmenta illius [82] axis, qui exprimit distantias, non expriment spatium confectum ; nam 
moveri debebit punctum utrumque : adhuc tamen ilia segmenta erunt proportionalia ipsi 
spatio confecto, eorum nimirum dimidio ; quod quidem satis est ad hoc, ut illae areae adhuc 
sint proportionales incrementis, vel decrementis quadrati velocitatum, adeoque ipsa 

178. Secundo loco notandum illud, ubi areae respondentes dato cuipiam spatio sint 
partim attractive, partim repulsivae, earum differentiam, quae oritur subtrahendo summam 
omnium repulsivarum a summa attractivarum, vel vice versa, exhibituram incrementum 
illud, vel decrementum quadrati velocitatis ; prout directio motus respectivi conspiret 
cum vi, vel oppositam habeat directionem. Quamobrem si interea, dum per aliquod majus 
intervallum a se invicem recesserunt puncta, habuerint vires directionis utriusque ; ut 
innotescat, an celeritas creverit, an decreverit & quantum ; erit investigandum, an areas 
omnes attractivae simul, omnes repulsivas simul superent, an deficiant, & quantum ; inde 
enim, & a velocitate, quae habebatur initio, erui poterit quod quaeritur. 

^ e arcubus, & areis ; nunc aliquanto diligentius considerabimus 
tangentis: sectio- ilia axis puncta, ad quae curva appellit. Ea puncta vel sunt ejusmodi, ut in iis curva axem 
ducT enera UmltUm secet > cu jusmodi in fig. I sunt E,G,I, &c., vel ejusmodi, ut in iis ipsa curva axem contingat 
tantummodo. Primi generis puncta sunt ea, in quibus fit transitus a repulsionibus ad 
attractiones, vel vice versa, & hsec ego appello limites, quod nimirum sint inter eas opposi- 
tarum directionum vires. Sunt autem hi limites duplicis generis : in aliis, aucta distantia, 
transitur a repulsione ad attractionem : in aliis contra ab attractione ad repulsionem. 
Prioris generis sunt E,I,N,R ; posterioris G,L,P : & quoniam, posteaquam ex parte 
repulsiva in una sectione curva transiit ad partem attractivam ; in proxime sequent! sectione 
debet necessario ex parte attractiva transire ad repulsivam, ac vice versa ; patet, limites 
fore alternatim prioris illius, & hujus posterioris generis. 

t P rro linrites prioris generis, a limitibus posterioris ingens habent inter se dis- 
differant': limites crimen. Habent illi quidem hoc commune, ut duo puncta collocata in distantia unius 
cohaesionls' & n n h' 111 ^ 8 cujuscunque nullam habeant mutuam vim, adeoque si respective quiescebant, pergant 
itidem respective quiescere. At si ab ilia respectiva quiete dimoveantur ; turn vero in 
limite primi generis ulteriori dimotioni resistent, & conabuntur priorem distantiam recu- 
perare, ac sibi relicta ad illam ibunt ; in limite vero secundi generis, utcunque parum 
dimota, sponte magis fugient, ac a priore distantia statim recedent adhuc magis. Nam 
si distantia minuatur ; habebunt in limite prioris generis vim repulsivam, quae obstabit 
uteriori accessui, & urgebit puncta ad mutuum recessum, quern sibi relicta acquirent, [83] 
adeoque tendent ad illam priorem distantiam : at in limite secundi generis habebunt 
attractionem, qua adhuc magis ad se accedent, adeoque ab ilia priore distantia, quae erat 
major, adhuc magis sponte fugient. Pariter si distantia augeatur, in primo limitum genere 
a vi attractiva, quse habetur statim in distantia majore ; habebitur resistentia ad ulteriorem 
recessum, & conatus ad minuendam distantiam, ad quam recuperandam sibi relicta tendent 
per accessum ; at in limitibus secundi generis orietur repulsio, qua sponte se magis adhuc 
fugient, adeoque a minore ilia priore distantia sponte magis recedent. Hinc illos prioris 
generis limites, qui mutuse positionis tenaces sunt, ego quidem appellavi limites coh&sionis, 
& secundi generis limites appellavi limites non cobasionis. 

Duo genera 

181. Ilia puncta, in quibus curva axem tangit, sunt quidem terminus quidam virium, 
quae ex utraque parte, dum ad ea acceditur, decrescunt ultra quoscunque limites, ac demum 
ibidem evanescunt ; sed in iis non transitur ab una virium directione ad aliam. Si con- 
tactus fiat ab arcu repulsive ; repulsiones evanescunt, sed post contactum remanent itidem 
repulsiones ; ac si ab arcu attractive, attractionibus evanescentibus attractiones iterum 
immediate succedunt. Duo puncta collocata in ejusmodi distantia respective quiescunt ; 



177. However, there are here two things that want noting only. The first of them The same result 
is this, that if two points approach one another or recede from one another in the straight holds good even 

,...., , r i ,. , ,. i when the segments 

line joining them, the segments of the axis, which expresses distances, do not represent of the axis are the 
the distances traversed ; for both points will have to move. Nevertheless the segments ' ialves of the d is- 

11 -11 i i i T i i i if f i . . i , tances traversed by 

will still be proportional to the distance traversed, namely, the half of it ; & this indeed is single points, 
sufficient for the areas to be still proportional to the increments or decrements of the 
squares of the velocities, & thus to represent them. 

178. In the second place it is to be noted that, where the areas corresponding to any if the areas are 
given interval are partly attractive & partly repulsive, their difference, obtained by p^ti* tt 2SS2 & 
subtracting the sum of all those that are repulsive from the sum of those that are attractive, their difference 
or vice versa, will represent the increment, or the decrement, of the square of the velocity, must be taken - 
according as the direction of relative motion is in the same direction as the force, or in 

the opposite direction. Hence, if, during the time that the points have receded from 
one another by some considerable interval, they had forces in each direction ; then 
in order to ascertain whether the velocity had been increased or decreased, & by how 
much, it will have to be considered whether all the attractive areas taken together are 
greater or less than all the repulsive areas taken together, & by how much. For from this, 
& from the velocity which initially existed, it will be possible to deduce what is required. 

179. So much for the arcs & the areas; now we must consider in a rather more careful Approach of the 
manner those points of the axis to which the curve approaches. These points are either ^en it cSa^or 
such that the curve cuts the axis in them, for instance, the points E, G, I, &c. in Fig. I : touches it; two 
or such that the curve only touches the axis at the points. Points of the first kind are u^ns^/'ihnit- 
those in which there is a transition from repulsions to attractions, or vice-versa ; & these points. 

I call limit-points or boundaries, since indeed they are boundaries between the forces acting 
in opposite directions. Moreover these limit-points are twofold in kind ; in some, when 
the distance is increased, there is a transition from repulsion to attraction ; in others, on 
the contrary, there is a transition from attraction to repulsion. The points E, I, N, R 
are of the first kind, and G, L, P are of the second kind. Now, since at one intersection, 
the curve passes from the repulsive part to the attractive part, at the next following 
intersection it is bound to pass from the attractive to the repulsive part, & vice versa. 
It is clear then that the limit-points will be alternately of the first & second kinds. 

1 80. Further, there is a distinction between limit-points of the first & those of the in what they agree 
second kind. The former kind have this property in common ; namely that, if two points * iff j . w ^ u *? t y 
are situated at a distance from one another equal to the distance of any one of these limit- points of cohesion 
points from the origin, they will have no mutual force ; & thus, if they are relatively & of non - cohesic i. 
at rest with regard to one another, they will continue to be relatively at rest. Also, if 

they are moved apart from this position of relative rest, then, for a limit-point of the first 
kind, they will resist further separation & will strive to recover the original distance, & 
will attain to it if left to themselves ; but, in a limit-point of the second kind, however 
small the separation, they will of themselves seek to get away from one another & will 
immediately depart from the original distance still more. For, if the distance is diminished, 
they will have, in a limit-point of the first kind, a repulsive force, which will impede further 
approach & impel the points to mutual recession, & this they will acquire if left to 
themselves ; thus they will endeavour to maintain the original distance apart. But in a 
limit-point of the second kind they will have an attraction, on account of which they will 
approach one another still more ; & thus they will seek to depart still further from the 
original distance, which was a greater one. Similarly, if the distance is increased, in 
limit-points of the first kind, due to the attractive force which is immediately obtained 
at this greater distance, there will be a resistance to further recession, & an endeavour 
to diminish the distance ; & they will seek to recover the original distance if left to 
themselves by approaching one another. But, in limit-points of the second class, a repulsion 
is produced, owing to which they try to get away from one another still further ; & thus 
of themselves they will depart still more from the original distance, which was less. On 
this account indeed I have called those limit-points of the first kind, which are tenacious 
of mutual position, limit-points of cohesion, & I have termed limit-points of the second 
kind limit-points of non-cohesion. 

181. Those points in which the curve touches the axis are indeed end-terms of series Two kinds of con- 
of forces, which decrease on both sides, as approach to these points takes place, beyond tactt 

all limits, & at length vanish there ; but with such points there is no transition from 
one direction of the forces to the other. If contact takes place with a repulsive arc, the 
repulsion vanishes, but after contact remains still a repulsion. If it takes place with an 
attractive arc, attraction follows on immediately after a vanishing attraction. Two points 
situated such a distance remain in a state of relative rest ; but in the first case they will 



pro forma curvae 
prope sectionem. 

sed in prime casu resistunt soli compressioni, non etiam distractioni, in secundo resistunt 
huic soli, non illi. 

l ^ 2 ' Limites cohsesionis possunt esse validissimi, & languidissimi. Si curva ibi quasi 
ad pcrpendiculum secat axem, & ab eo longissime recedit ; sunt validissimi : si autem 
ip Sum secet in angulo perquam exiguo, & parum ab ipso recedat ; erunt languidissimi. 
Primum genus limitum cohsesionis exhibet in fig. I arcus tNy, secundum cNx. In illo 
assumptis in axe Nz, NM utcunque exiguis, possunt vires zt, uy, & areae Nzt, Nwy esse 
utcumque magnas, adeoque, mutatis utcunque parum distantiis, possunt haberi vires ab 
ordinatis expressae utcunque magnae, quae vi comprimenti, vel distrahenti, quantum libuerit, 
valide resistant, vel areae utcunque magnae, quae velocitates quantumlibet magnas 
respectivas elidant, adeoque sensibilis mutatio positionis mutuae impediri potest contra 
utcunque magnam vel vim prementem, vel celeritatem ab aliorum punctorum actionibus 
impressam. In hoc secundo genere limitum cohaesionis, assumptis etiam majoribus 
segmentis Nz, Nw, possunt & vires zc,ux, & areae Nzf , Ntf, esse quantum libuerit exiguae, 
& idcirco exigua itidem, quantum libuerit, resistentia, quae mutationem vetet. 

P ssunt autem hi Hmites esse quocunque, utcunque magno numero ; cum 
ro, utcunque proxi- demonstratum sit, posse curvam in quotcunque, & quibuscunque punctis axem secare. 
mos, vel remotes p oss unt idcirco etiam esse utcunque inter se proximi, vel remoti, ut [84] alicubi intervallum 
originis' abscissa- inter duos proximos limites sit etiam in quacunque ratione majus, quam sit distantia 
ordme praecedentis ab origine abscissarum A ; alibi in intervallo vel exiguo, vel ingenti sint quam- 
plurimi inter se ita proximi, ut a se invicem distent minus, quam pro quovis assumpto, 
aut dato intervallo. Id evidenter fluit ex eo ipso, quod possint sectiones curvae cum axe 
haberi quotcunque, & ubicunque. Sed ex eo, quod arcus curvae ubicunque possint habere 
positiones quascunque, cum ad datas curvas accedere possint, quantum libuerit, sequitur, 
quod limites ipsi cohaesionis possint alii aliis esse utcunque validiores, vel languidiores, 
atque id quocunque ordine, vel sine ordine ullo ; ut nimirum etiam sint in minoribus 
distantiis alicubi limites validissimi, turn in majoribus languidiores, deinde itidem in 
majoribus multo validiores, & ita porro ; cum nimirum nullus sit nexus necessarius inter 
distantiam limitis ab origine abscissarum, & ejus validitatem pendentem ab inclinatione, 
& recessu arcus secantis respectu axis, quod probe notandum est, futurum nimirum usui 
ad ostendendum, tenacitatem, sive cohaesionem, a densitate non pendere. 


Quse positio rectae jg^.. I n utroque limitum genere fieri potest, ut curva in ipso occursu cum axe pro 

infinite 3 rarissima! tangente habeat axem ipsum, ut habeat ordinatam, ut aliam rectam aliquam inclinatam. 
quae frequentissima. J n primo casu maxime ad axem accedit, & initio saltern languidissimus est limes ; in secundo 
maxime recedit, & initio saltern est validissimus ; sed hi casus debent esse rarissimi, si 
uspiam sunt : nam cum ibi debeat & axem secare curva, & progredi, adeoque secari in 
puncto eodem ab ordinata producta, debebit habere flexum contrarium, sive mutare 
directionem flexus, quod utique fit, ubi curva & rectam tangit simul, & secat. Rarissimos 
tamen debere esse ibi hos flexus, vel potius nullos, constat ex eo, quod flexus contrarii puncta 
in quovis finito arcu datae curvae cujusvis numero finite esse debent, ut in Theoria curvarum 
demonstrari potest, & alia puncta sunt infinita numero, adeoque ilia cadere in intersectiones 
est infinities improbabilius. Possunt tamen saepe cadere prope limites : nam in singulis 
contorsionibus curvae saltern singuli flexus contrarii esse debent. Porro quamcunque 
directionem habuerit tangens, si accipiatur exiguus arcus hinc, & inde a limite, vel 
maxime accedet ad rectam, vel habebit curvaturam ad sensum aequalem, & ad sensum 
aequali lege progredientem utrinque, adeoque vires in aequali distantia exigua a limite 
erunt ad sensum hinc, & inde aequales ; sed distantiis auctis poterunt & diu aequalitatem 
retinere, & cito etiam ab ea recedere. 

Transitus per infi- 185. Hi quidem sunt limites per intersectionem curvae cum axe, viribus evanescentibus 

as tl m"toticis ribUS m *P SO li m i te - At possunt [85] esse alii limites, ac transitus ab una directione virium ad 

aliam non per evanescentiam, sed per vires auctas in infinitum, nimirum per asymptoticos 


resist compression only, & not separation ; and in the second case the latter only, but not 
the former. 

182. Limit-points may be either very strong or very weak. If the curve cuts the axis The limit-points of 
at the point almost at right angles, & goes off to a considerable distance from it, they o h ^eak ?ccordf 
are very strong. But if it cuts the axis at a very small angle & recedes from it but little, to the form of the 
then they will be very weak. The arc *Ny in Fig. i represents the first kind of limit- ^Hint * iVater- 
points of cohesion, and the arc cNx the second kind. At the point N, if Nz, N are section. 

taken along the axis, no matter how small, the forces zt, uy, & the areas Nzt, Ny may 
be of any size whatever ; & thus, if the distances are changed ever so little, it is possible 
that there will be forces represented by ordinates ever so great ; & these will strongly 
resist the compressing or separating force, be it as great as you please ; also that we shall 
have areas, ever so large, that will destroy the relative velocities, no matter how great they 
may be. Thus, a sensible change of relative position will be hindered in opposition to 
any impressed force, however great, or against a velocity generated by the actions upon 
them of other points. In the second kind of limit-points of cohesion, if also segments Nz, 
Nw are taken of considerable size even, then it is possible for both the forces zc, ux, & 
the areas Nzc, Nux to be as small as you please ; & therefore also the resistance that 
opposes the change will be as small as you please. 

183. Moreover, there can be any number of these limit-points, no matter how great ; The limit-points 
for it has been proved that the curve can cut the axis in any number of points, & anywhere. are m <J efimte as 

rrM F i i r i i i i f i i 6 g cl i Cl S HUTU DCr, 

I herefore it is possible for them to be either close to or remote from one another, without their proximity to 

any restriction whatever, so that the interval between any two consecutive limit-points one^another^&^th^ 

at any place shall even bear to the distance of the first of the two from A, the origin of order of their occur- 

abscissae, a ratio that is greater than unity. In other words, in any interval, either very ^toe <SgVof P ab- 

small or very large, there may be an exceedingly large number of them so close to one scissae. 

another, that they are less distant from one another than they are from any chosen or given 

interval. This evidently follows from the fact that the intersections of the curve with 

the axis can happen any number of times & anywhere. Again, from the fact that arcs 

of the curve can anywhere, owing to their being capable of approximating as closely as 

you please to given curves, have any positions whatever, it follows that these limit-points 

of cohesion can be some of them stronger than others, or weaker, in any manner ; & 

that too, in any order, or without order. So that, for instance, we may have at small 

distances anywhere very strong limit-points, then at greater distances weaker ones, & 

then again at still greater distances much stronger ones, & so on. That is to say, since 

there is no necessary connection between the distance of a limit-point from the origin of 

abscissae and its strength, which depends on the inclination of the intersecting arc & the 

distance it recedes from the axis. It is well that this should be made a note of ; for indeed 

it will be used later to prove that tenacity or cohesion does not depend on density. 

184. In each of these kinds of limit-points it may happen that the curve, where it What position of 
meets the axis, may have the axis itself as its tangent, or the ordinate, or any other straight touchkig^he* curve 
line inclined to the axis. In the first case it approximates very closely to the axis, & at a limit-point is 
close to the point at any rate it is a very weak limit-point ; in the second case, it departs ^at magt^fr* 
from the axis very sharply, & close to the point at any rate it is a very strong limit-point, quent ; small arcs 
But these two cases must be of very rare occurrence, if indeed they ever occur. For, since fTm^tVo'in t* are 
at the point the curve is bound to cut the axis & go on, & thus be cut in the same point equal & similar, 
by the ordinate produced, it is bound to have contrary-flexure ; that is to say, a change 

in the direction of its curvature, such as always takes place at a point where the curve both 
touches a straight line & cuts it at the same time. Yet, that these flexures must occur 
very rarely at such points, or rather never occur at all, is evident from the fact that in any 
finite arc of any given curve the number of points of contrary-flexure must be finite, as can 
be proved in the theory of curves ; & other points are infinite in number ; hence that the 
former should happen at the points of intersection with the axis is infinitely improbable. 
On the other hand they may often fall close to the limit-points ; for in each winding of 
the curve about the axis there must be at least one point of contrary-flexure. Further, 
whatever the direction of the tangent, if a very small arc of the curve is taken on each side 
of the limit-point, this arc will either approximate very closely to the straight line, or will 
have its curvature the same very nearly, & will proceed very nearly according to the same 
law on each side ; & thus the forces, at equal small distances on each side of the limit- 
point will be very nearly equal to one another ; but when the distances are increased, 
they can either maintain this equality, for some considerable time, or indeed very soon 
depart from it. 

185. The limit-points so far discussed are those obtained through the intersection Passage through 
of the curve with the axis, where the forces vanish at the limit-point. But there 

may be other limit-points ; the transition from one direction of the forces to another 




curvse arcus. Diximus supra num. 168. adnot. (i), quando crus asymptoticum abit in 
infinitum, debere ex infinite regredi crus aliud habens pro asymptote eandem rectam, & 
posse regredi cum quatuor diversis positionibus pendentibus a binis partibus ipsius rectae, 
& binis plagis pro singulis rectae partibus ; sed cum nostra curva debeat semper progredi, 
diximus, relinqui pro ea binas ex ejusmodi quatuor positionibus pro quovis crure abeunte 
in infinitum, in quibus nimirum regressus fiat ex plaga opposita. Quoniam vero, progre- 
diente curva, abire potest in infinitum tarn crus repulsivum, quam crus attractivum , jam 
iterum fiunt casus quatuor possibiles, quos exprimunt figurae 16, 17, 18, & 19, in quibus 
omnibus est axis ACS, asymptotus DCD', crus recedens in infinitum EKF, regrediens 
ex infinite GMH. 



I C 


FIG. 1 6. 

I C 




FIG. 17. 

Quatuor eorum 186. In fig. 16. cruri repulsivo EKF succedit itidem repulsivum GMH ; in fig. 17 

f^nXntesTontac- repulsivo attractivum ; in 1 8 attractive attractivum; in 19 attractive repulsivum. Primus 

tibus, bini Hmiti- & tertius casus respondent contactibus. Ut enim in illis evanescebat vis ; sed directionem 

ionU, al aHer COh non non mutabat ; ita & hie abit quidem in infinitum, sed directionem non mutat. Repulsioni 

cohaesionis. IK in fig. 1 6 succedit repulsio LM ; & attractioni in fig. 18 attractio. Quare ii casus non 

habent limites quosdam. Secundus, & quartus habent utique limites ; nam in fig. 17 

repulsion! IK succedit attractio LM ; & in fig. 19 attractioni repulsio ; atque idcirco 

secundus continet limitem cobasionis, quartus limitem non cohcesionis. 

Nuiium in Natura 

vero eum 

187. Ex istis casibus a nostra curva censeo removendos esse omnes praeter solum 
quartum ; & in hoc ipso removenda omnia crura, in quibus ordinata crescit in ratione 
ipsum minus, quam simplici reciproca distantiarum a limite. Ratio excludendi est, ne haberi 
aliquando vis infinita possit, quam & per se se absurdam censeo, & idcirco praeterea, quod 
infinita vis natura sua velocitatem infinitam requirit a se generandam finito tempore. Nam 
in primo, & secundo casu punctum collocatum in ea distantia ab alio puncto, quam habet 
I, ab origine abscissarum, abiret ad C per omnes gradus virium auctarum in infinitum, 
& in C deberet habere vim infinitam ; in tertio vero idem accideret puncto collocate in 
distantia, quam habet L. At in quarto casu accessum ad C prohibet ex parte I attractio 
IK, & ex parte L repulsio LM. Sed quoniam, si eae crescant in ratione reciproca minus, 
quam simplici distantiarum CI,CL ; area FKICD, vel GMLCD erit finita, adeoque 
punctum impulsum versus C velocitate majore, quam quae respondeat illi areae, debet 
transire per omnes virium magnitudines usque ad vim absolute infinitam in C, quae ibi 
[86] praeterea & attractiva esse deberet, & repulsiva, limes videlicet omnium & attracti- 
varum, & repulsivarum ; idcirco ne hie quidem casus admitti debet, nisi cum hac 
conditione, ut ordinata crescat in ratione reciproca simplici distantiarum a C, vel etiam 
majore, ut nimirum area infinita evadat, & accessum a puncto C prohibeat. 


may occur, not with evanescence of the forces, but through the forces increasing indefinitely, 
that is to say through asymptotic arcs of the curve. We said above, in Note () to Art. 
1 68, when an asymptotic arm goes off to infinity, there must be another asymptotic arm 
returning from infinity having the same straight line for an asymptote ; & it may return 
in four different positions, which depend on the two parts of the straight line & the two 
sides of each part of the straight line. But, since our curve must always go forward, we 
said that for it there remained only two out of these four positions, for any arm going off 
to infinity ; that is to say, those in which the return is made on the opposite side of the 
straight line. However, since, whilst the curve goes forward, either a repulsive or an 
attractive arm can go off to infinity, here again we must have four possible cases, represented 
in Figs. 16, 17, 18, 19, in all of which ACB is the axis, DCD' the asymptote, EKF the 
arm going off to infinity, & GMH the arm returning from infinity. 

I C 


n cr 

FIG 19. 

1 86. In Fig. 1 6, to a repulsive arm EKF there succeeds an arm that is also repulsive; 
in Fig. 17, to a repulsive succeeds an attractive ; in Fig. 18, to an attractive succeeds an 
attractive ; and in Fig. 19, to an attractive succeeds a repulsive. The first & third cases 
correspond to contacts. For, just as in contact, the force vanished, but did not change 
its direction, so here also the force indeed becomes infinite but does not change its direction. 
In Fig. 1 6, to the repulsion IK there succeeds the repulsion LM, & in Fig. 18 to an 
attraction an attraction ; & thus these two cases cannot have any limit-points. But 
the second & fourth cases certainly have limit-points ; for, in Fig. 17, to the repulsion 
IK there succeeds the attraction LM, & in Fig. 19 to an attraction a repulsion ; & 
thus the second case contains a limit-point of cohesion, & the fourth a limit-point of 

187. Out of these cases I think that all except the last must be barred from our curve ; 
& even with that all arms must be rejected for which the ordinates increase in a ratio 
less than the simple reciprocal of the distances from the limit-point. My reasons for 
excluding these are to avoid the possibility of there being at any time an infinite force 
(which of itself I consider to be impossible), & because, in addition to that, an infinite 
force, by its very nature necessitates the creation by it of an infinite velocity in a finite time. 
For, in the first & second cases, a point, situated at the distance from another point equal 
to that which I has from the origin of abscissae, would go off to C through all stages of 
forces increased indefinitely, & at C would be bound to have an infinite force. In the 
third case, too, the same thing would happen to a point situated at a distance equal to that 
of L. Now, in the fourth case, the approach to C is restrained, from the side of I by the 
attraction IK, & from the side of L by the repulsion LM. However, since, if these 
forces increase in a ratio that is less than the simple reciprocal ratio of the distances CI, 
CL, then the area FKICD or the area GMLCD will be finite ; thus the point, being 
impelled towards C with a velocity that is greater than that corresponding to the area, 
must pass through all magnitudes of the forces up to a force that is absolutely infinite at 
C ; and this force must besides be both attractive & repulsive, the limit so to speak of all 
attractive & repulsive forces. Hence not even this case is admissible, unless with the 
condition that the ordinate increases in the simple reciprocal ratio of the distances from C, 
or in a greater ; that is to say, the area must turn out to be infinite and so restrain the 
approach towards the point C. 

Four kinds of 
them ; two corre- 
sponding to contact, 
& two to limit- 
points, of which the 
one is a limit-point 
of cohesion & the 
other of non-cohe- 

None of these ex- 
cept the last admis- 
sible in Nature ; & 
not even that in 


Transitus per eum 188. Quando habeatur hie quartus casus in nostra curva cum ea conditione ; turn 

bills: teqaibai quidem nullum punctum collocatum ex alters parte puncti C poterit ad alteram transilire, 
distantiis constet, quacunque velocitate ad accessum impellatur versus alterum punctum, vel ad recessum 

eum non haben. u j i i TJ 

ab ipso, impediente transitum area repulsiva mnnita, vel innnita attractiva. Inde vero 
facile colligitur, eum casum non haberi saltern in ea distantia, quae a diametris minimarum 
particularum conspicuarum per microscopia ad maxima protenditur fixarum intervalla 
nobis conspicuarum per telescopia : lux enim liberrime permeat intervallum id omne. 
Quamobrem si ejusmodi limites asymptotici sunt uspiam, debent esse extra nostrae sensibi- 
litatis sphaeram, vel ultra omnes telescopicas fixas, vel citra microscopicas moleculas. 

Transitus ad puncta 189. Expositas hisce, quae ad curva virium pertinebant, aggrediar simpliciora quaadam, 

iae, & massas. maxime notatu digna sunt, ac pertinent ad combinationem punctorum primo quidem 

duorum, turn trium, ac deinde plurium in massa etiam coalescentium, ubi & vires mutuas, 
& motus quosdam, & vires, quas in alia exercent puncta, considerabimus. 

in limitibus ; icp. Duo puncta posita in distantia aequali distantiae limitis cujuscunque ab origine 

? 1 ^ ' abscissarum, ut in fig. i. AE, AG, AI, &c, (immo etiam si curva alicubi axem tangat, aequali 
distantiae contactus ab eodem), ac ibi posita sine ulla velocitate, quiescent, ut patet, quia 
nullam habebunt ibi vim mutuam : posita vero extra ejusmodi limites, incipient statim 
ad se invicem accedere, vel a se invicem recedere per intervalla aequalia, prout fuerint sub 
arcu attractivo, vel repulsive. Quoniam autem vis manebit semper usque ad proximum 
limitem directionis ejusdem ; pergent progredi in ea recta, quae ipsa urgebat prius, usque 
ad distantiam limitis proximi, motu semper accelerate, juxta legem expositam num. 176, 
ut nimirum quadrata velocitatum integrarum, quae acquisitae jam sunt usque ad quodvis 
momentum (nam velocitas initio ponitur nulla) respondeant areis clausis inter ordinatam 
respondentem puncto axis terminanti abscissam, quae exprimebat distantiam initio motus, 
& ordinatam respondentem puncto axis terminanti abscissam, quae exprimit distantiam 
pro eo sequent! momento. Atque id quidem, licet interea occurrat contactus aliquis ; 
quamvis enim in eo vis sit nulla, tamen superata distantia per velocitatem jam acquisitam, 
statim habentur iterum [87] vires ejusdem directionis, quae habebatur prius, adeoque 
perget acceleratio prioris motus. 

Motus post proxi- l q I p rO ximus limes erit ems generis, cuius generis diximus limites cohaesioms, in quo 

mum limitem super- . . 7 . ,. . , J . i j ^ i 

atum, & osciiiatio. nimirum si distantia per repulsionem augebatur, succedet attractio ; si vero minuebatur 
per attractionem, succedet e contrario repulsio, adeoque in utroque casu limes erit ejusmodi, 
ut in distantiis minoribus repulsionem, in majoribus attractionem secum ferat. In eo 
limite in utroque casu recessus mutui, vel accessus ex praecedentibus viribus, incipiet, 
velocitas motus minui vi contraria priori, sed motus in eadem directione perget ; donee 
sub sequent! arcu obtineatur area curvae aequalis illi, quam habebat prior arcus ab initio 
motus usque ad limitem ipsum. Si ejusmodi aequalitas obtineatur alicubi sub arcu 
sequente ; ibi, extincta omni praecedenti velocitate, utrumque punctum retro reflectet 
cursum ; & si prius accedebant, incipient a se invicem recedere ; si recedebant, incipient 
accedere, atque id recuperando per eosdem gradus velocitates, quas amiserant, usque ad 
limitem, quern fuerant prsetergressa ; turn amittendo, quas acquisiverant usque ad dis- 
tantiam, quam habuerant initio ; viribus nimirum iisdem occurrentibus in ingressu, & 
areolis curvae iisdem per singula tempuscula exhibentibus quadratorum velocitatis incre- 
menta, vel decrementa eadem, quae fuerant antea decrementa, vel incrementa. Ibi autem 
iterum retro cursum reflectent, & oscillabunt circa ilium cohaesionis limitem, quern fuerant 
praetergressa, quod facient hinc, & inde perpetuo, nisi aliorum externorum punctorum 
viribus perturbentur, habentia velocitatem maximam in plagam utramlibet in distantia 
ipsius illius limitis cohaesionis. 


osdiiationis jo.2. Quod si ubi primum transgressa sunt proximum limitem cohaesionis, offendant 

' S arcum ita minus validum praecedente, qui arcus nimirum ita minorem concludat aream, 

quam praecedens, ut tota ejus area sit aequalis, vel etiam minor, quam ilia praecedentis 

arcus area, quae habetur ab ordinata respondente distantiae habitse initio motus, usque ad 


1 88. When, if ever, this fourth case occurs in our curve, then indeed no point situated Passage through a 
on either side of the point C will be able to pass through it to the other side, no matter L^dTslmpo'ssibiel 
what the velocity with which it is impelled to approach towards, or recede from, the other distances at which 
point ; for the infinite repulsive area, or the infinite attractive area, will prevent such $, e re are^iio such 
passage. Now, it can easily be derived from this, that this case cannot happen at any rate limit-points. 

in the distance lying between the diameters of the smallest particles visible under the 
microscope & the greatest distances of the stars visible to us through the telescope ; for 
light passes with the greatest freedom through the whole of this interval. Therefore, if 
there are ever any such asymptotic limit-points, they must be beyond the scope of our 
senses, either superior to all telescopic stars, or inferior to microscopic molecules. 

1 89. Having thus set forth these matters relating to the curve of forces, I will now w ? n w pass on to 
discuss some of the simpler things that are more especially worth mentioning with regard ^^ s . of matter> & 
to combination of points ; & first of all I will consider a combination of two points, then 

of three, & then of many, coalescing into masses ; & with them we will discuss their 
mutual forces, & certain motions, and forces, which they exercise on other points. 

190. Two points situated at a distance apart equal to the distance of any limit-point Rest at Hmit- 
from the origin of abscissae, like AE, AG, AI, &c. in Fig. I (or indeed also where the r 

curve touches the axis anywhere, equal to the distance of the point of contact from the without them. 

origin), & placed in that position without any velocity, will be relatively at rest ; this is 

evident from the fact that they have then no mutual force ; but if they are placed at any 

other distance, they will immediately commence to move towards one another or away 

from one another through equal intervals, according as they lie below an attractive or a 

repulsive arc. Moreover, as the force always remains the same in direction as far as the 

next following limit-point, they continue to move in the same straight line which contained 

them initially as far as the distance apart equal to the distance of the next limit-point 

from the origin, with a motion that is continually accelerated according to the law given 

in Art. 176 ; that is to say, in such a manner that the squares of the whole velocities which 

have been already acquired up to any instant (for the velocity at the commencement is 

supposed to be nothing) will correspond to the areas included between the ordinate 

corresponding to the point of the axis terminating the abscissa which the distance traversed 

since motion began and the ordinate corresponding to the point on the axis terminating 

the abscissa which expresses the distance for the next instant after it. This is still the case, 

even if a contact should occur in the meantime. For, although at a point where contact 

occurs the force is nothing, yet, this distance being passed by the velocity already acquired, 

immediately afterwards there will be forces having the same direction as before ; and thus 

the acceleration of the former motion will proceed. 

191. The next limit-point will be one of the kind we have called limit-points of cohesion, Motion after the 
namely, one in which, if the distance is increased by repulsion, then attraction follows ; passed^osc^Sion 5 
but if the distance is diminished by attraction, then on the contrary repulsion will follow ; 

& thus, in either case, the limit-point will be of such a kind, that it gives a repulsion at 
smaller distances & an attraction at larger. In this limit-point, in either case, the separation 
or approach, due to the forces that have preceded, will be changed, & the velocity of motion 
will begin to be diminished by a force opposite to the original force, but the motion will 
continue in the same direction ; until an area of the curve under the arc that follows the 
limit-point becomes equal to the area under the former arc from the commencement of 
the motion as far as the limit-point. If equality of this kind is obtained somewhere under 
the subsequent arc, then, the whole of the preceding velocity being destroyed, both the 
points will return along their paths ; & if at the start they approached one another, they 
will now begin to recede from one another, or if they originally receded from one another, 
they will now commence to approach ; and as they do this, they will regain by the same 
stages the velocities which they lost, as far as the limit-point which they passed through ; 
then they will lose those which they had acquired, until they reach the distance 
apart which they had at the commencement. That is to say, the same forces occur on 
the return path, & the same little areas of the curve for the several short intervals of time 
represent increments or decrements of the squares of the velocities which are the same 
as were formerly decrements or increments. Then again they will once more retrace their 
paths, & they will oscillate about the limit-point of cohesion which they had passed through ; 
& this they will do, first on this side & then on that, over & over again, unless they are disturbed 
by forces due to other points outside them ; & their greatest velocity in either direction 
will occur at a distance apart equal to that of the limit-point of cohesion from the origin. 

192. But if, when they first passed through the nearest limit-point of cohesion, they The case of a larger 
happened to come to an arc representing forces so much weaker than those of the preceding 

ti-ii f , 11 i ft T 

arc that the whole area of it was equal to, or even less than, the area of the preceding arc, 
reckoning from the ordinate corresponding to the distance apart at the commencement 


limitem ipsum ; turn vero devenient ad distantiam alterius limitis proximi priori, qui 
idcirco erit limes non cohaesionis. Atque ibi quidem in casu sequalitatis illarum arearum 
consistent, velocitatibus prioribus elisis, & nulla vi gignente novas. At in casu, quo tota 
ilia area sequentis arcus fuerit minor, quam ilia pars areae praecedentis, appellent ad dis- 
tantiam ejus limitis motu quidem retardate, sed cum aliqua velocitate residua, quam 
distantiam idcirco praetergressa, & nacta vires directionis mutatse jam conspirantes cum 
directione sui motus, non, ut ante, oppositas, accelerabunt motum usque ad distantiam 
limitis proxime sequentis, quam praetergressa precedent, sed motu retardato, ut in priore ; 
& si area sequentis arcus non sit par extinguendae ante suum finem toti [88] velocitati, 
quae fuerat residua in appulsu ad distantiam limitis praecedentis non cohaesionis, & quae 
acquisita est in arcu sequent! usque ad limitem cohsesionis proximum ; turn puncta 
appellent ad distantiam limitis non cohaesionis sequentis, ac vel ibi sistent, vel progredientur 
itidem, eritque semper reciprocatio quaedam motus perpetuo accelerati, turn retardati ; 
donee deveniatur ad arcum ita validum, nimirum qui concludat ejusmodi aream, ut tota 
velocitas acquisita extinguatur : quod si accidat alicubi, & non accidat in distantia alicujus 
limitis ; cursum reflectent retro ipsa puncta, & oscillabunt perpetuo. 

Velocitatis muta- 193. Porro in hujusmodi motu patet illud, dum itur a distantia limitis cohaesionis 

^"^abeat^maxU a ^ distantiam limitis non cohaesionis, velocitatem semper debere augeri ; turn post 

mum, & minimum transitum per ipsam debere minui, usque ad appulsum ad distantiam limitis non cohaesionis, 

extmgui possit a d eO q ue habebitur semper in ipsa velocitate aliquod maximum in appulsu ad distantiam 

limitis cohaesionis, & minimum in appulsu ad distantiam limitis non cohaesionis. Quamo- 

brem poterit quidem sisti motus in distantia limitis hujus secundi generis ; si sola existant 

ilia duo puncta, nee ullum externum punctum turbet illorum motum : sed non poterit 

sisti in distantia limitis illius primi generis ; cum ad ejusmodi distantias deveniatur semper 

motu accelerate. Praeterea patet & illud, si ex quocunque loco impellantur velocitatibus 

aequalibus vel alterum versus alterum, vel ad partes oppositas, debere haberi reciprocationes 

easdem auctis semper aeque velocitatibus utriusque, dum itur versus distantiam limitis 

primi generis, & imminutis, dum itur versus distantiam limitis secundi generis. 

oscMatlo S ma/**? X 94' P atet & illud, si a distantia limitis primi generis dimoveantur vi aliqua, vel non 

esse debeat, & unde ita uigenti velocitate impressa, oscillationem fore perquam exiguam, saltern si quidam 

CJUS mag va h"dus fuerit limes ; nam velocitas incipiet statim minui, & ei vi statim vis contraria 
invenietur, ac puncta parum dimota a loco suo, turn sibi relicta statim retro cursum reflect- 
ent. At si dimoveantur a distantia limitis secundi generis vi utcunque exigua ; oscillatio 
erit multo major, quia necessario debebunt progredi ultra distantiam sequentis limitis 
primi generis, post quern motus primo retardari incipiet. Quin immo si arcus proximus 
hinc, & inde ab ejusmodi limite secundi generis concluserit aream ingentem, ac majorem 
pluribus sequentibus contrariae directionis, vel majorem excessu eorundem supra areas 
interjacentes directionis suae ; turn vero oscillatio poterit esse ingens : nam fieri poterit, 
ut transcurrantur hinc, & inde limites plurimi, antequam deveniatur ad arcum ita validum, 
ut velocitatem omnem elidat, & motum retro reflectat. Ingens itidem oscillatio esse 
poterit, si cum ingenti vi dimoveantur puncta a distantia limitum generis utriuslibet ; ac 
res tota pendet a velocitate initiali, & ab areis, quae post oc-[8Q]-currunt, & quadratum 
velocitatis vel augent, vel minuunt quantitate sibi proportionali. 

Accessum debere 195. Utcunque magna sit velocitas, qua dimoveantur a distantia limitum ilia duo 

swt^ saltern a^pmno p uncta> utcunque validos inveniant arcus conspirantes cum velocitatis directione, si ad 
recess um posse se invicem accedunt, debebunt utique alicubi motum retro reflectere, vel saltern sistere, 
cas^'^o^bms 1 saltern advenient ad distantias illas minimas, quae respondent arcui asymptotico, 
exiguae differentiae cujus area est capax cxtinguendse cujuscunque velocitatis utcunque magnae. At si 
velocitatis ingentis. rece( j ant a se mv i ce m, fieri potest, ut deveniant ad arcum aliquem repulsivum validissimum, 
cujus area sit major, quam omnis excessus sequentium arearum attractivarum supra repul- 


of the motion up to the limit-point ; then indeed they will arrive at a distance apart equal 
to that of the limit-point next following the first one, which will therefore be a limit-point of 
non-cohesion. Here they will stop, in the case of equality between the areas in question ; 
for the preceding velocities have been destroyed & no fresh ones will be generated. But 
in the case when the whole of the area under the second arc is less than the said part of 
the first area, they will reach a distance apart equal to that of the limit-point with a motion 
that is certainly diminished ; but some velocity will be left, & this distance will therefore 
be passed, & the points, coming under the influence of forces changed in direction so that they 
now act in the same sense as their own motion, will accelerate their motion as far as the 
next following limit-point ; & having passed through this they will go on, but with 
retarded motion as in the first case. Then, if the area of the subsequent arc is not capable 
before it ends of destroying the whole of the velocity which remained on attaining the 
distance of the preceding limit-point of non-cohesion, & that which was acquired in the 
arc that followed it up to the next limit-point of cohesion, then the points will move to a 
distance apart equal to that of the next following limit-point from the origin, & will either 
stop there or proceed ; & there will always be a repetition of the motion, continually 
accelerated & retarded. Until at length it comes to an arc so strong, that is to say, one 
under which the area is such, that the whole velocity acquired is destroyed ; & when this 
happens anywhere, & does not happen at a distance equal to that of any limit-point, then 
the points will retrace their paths & oscillate continuously. 

193. Further in this kind of motion it is clear that along the path from the distance Alternate changes 
of a limit-point of cohesion to a limit-point of non-cohesion the velocity is bound to be of velocity ; where 
always increasing ; then after passing through the latter it must decrease up to its arrival at the value! & ""a^mln? 
distance of a limit-point of non-cohesion. Thus, there will always be in the velocity a P um valu e ; where 
maximum on arrival at a distance equal to that of a limit-point of cohesion, & a minimum ' maybe estr yed. 
on arrival at a distance of a limit-point of non-cohesion. Hence indeed the motion may 

possibly cease at a limit-point of this second kind, if the two points exist by themselves, 
& no other point influences their motion from without. But it cannot cease at a distance 
of a limit-point of the first kind ; for it will always arrive at distances of this kind with 
an accelerated motion. Moreover it is also clear that, if they are urged from any given 
position with equal velocities, either towards one another or in opposite directions, the 
same alternations must be had as before, the velocities being increased equally for each 
point whilst they are moving up to a distance of a limit-point of the first kind, & diminished 
whilst they are moving up to a distance of a limit-point of the second kind. 

194. It is evident also that, if the points are moved from a distance apart equal to that of The limit-points 
a limit-point of the first kind by some force (especially when the velocity thus impressed oscfflation mus^be 
is not extremely great), then the oscillation will be exceedingly small, at least so long as the krger; & the thing 
limit-point is a fairly strong one. For the velocity will commence to be diminished tude" 
immediately, & to the force another force will be obtained at once, acting in opposition 

to it ; & the points, being moved but little from their original position, will immediately 
afterwards retrace their paths if left to themselves. But if they are moved from a distance 
apart equal to that of a limit-point of the second kind by any force, no matter how small, 
then the oscillation will be much greater ; for, of necessity, they are bound to go on beyond 
the distance equal to that of the next following limit-point of the first kind ; & not until 
this has been done, will the motion begin to be retarded. Nay, if the next arc on each 
side of such a limit-point of the second kind should include a very large area, and one that 
is greater than several of those subsequent to them, which are opposite in direction, or 
greater than the excess of these over the intervening areas that are in the same direction, 
then indeed the oscillation may be exceedingly large. For it may be that very many 
limit-points on either side are traversed before an arc is arrived at, which is sufficiently 
strong to destroy the whole of the velocity & reverse the direction of motion. A very 
large oscillation will also be possible, if the points are moved from a distance apart equal to 
that of a limit-point of either kind by an exceedingly large force. The whole thing depends 
on the initial velocity & the areas which occur subsequently, & either increase or decrease 
the square of the velocity by a quantity that is proportional to the areas themselves. 

195. However great the velocity may be, with which the two points are moved from Approach is bound 
a distance equal to that of any limit-point, no matter how strong are the arcs they come *^g S \* t t a h n e y % 
upon, which are in the same direction as that of the velocity ; yet, if they approach one repulsive arc, but 
another, they are bound somewhere to have their motion reversed, or at least to come on pa Sdennitety ; S a 
to rest ; for, at all events, they must finally attain to those very small distances that correspond noteworthy case 
to an asymptotic arm, the area of which is capable of destroying any velocity whatever, arSerencTfar a S very 
no matter how great. But, if they recede from one another, it may happen that they come great velocity. 

to some very strong repulsive arc, the area of which is greater than the whole of the excess 
of the subsequent attractive arcs above those that are repulsive, as far as the very weak 



sivas, usque ad languidissimum ilium arcum postremi cruris gravitatem exhibentis. Turn 
vero motus acquisitus ab illo arcu nunquam poterit a sequentibus sisti, & puncta ilia recedent 
a se invicem in immensum : quin immo si ille arcus repulsivus cum sequentibus repulsivis 
ingentem habeat areae excessum supra arcus sequentes attractivos ; cum ingenti velocitate 
pergent puncta in immensum recedere a se invicem ; & licet ad initium ejus tarn validi 
arcus repulsivi deveniant puncta cum velocitatibus non parum diversis ; tamen velocitates 
recessuum post novum ingens illud augmentum erunt parum admodum discrepantes a 
se invicem : nam si ingentis radicis quadrate addatur quadratum radicis multo minoris, 
quamvis non exiguae ; radix extracta ex summa parum admodum differet a radice priore. 

Demonstratio ad- 
modum simplex. 



FIG. 20. 

Quid accidat binis 
punctis, cum sunt 
sola, quid possit 
accidere actionibus 
aliorum externis. 

Si limites sint a se 
invicem r e m o t i, 
m u t a t a multum 
distantia r e d i ri 
retro : secus, si 
sint proximi. 

196. Id quidem ex Euclidea etiam Geometria manifestum fit. Sit in fig. 20 AB 
linea longior, cui addatur ad perpendiculum BC, multo minor, quam fit ipsa ; turn centre A, 
intervallo AC, fiat semicirculus occurrens AB hinc, & inde in E, D. Quadrate AB 
addendo quadratum BC habetur quadratum AC, sive AD ; & tamen haec excedit prsece- 
dentem radicem AB per solam BD, quae semper est minor, quam BC, & est ad ipsam, ut 
est ipsa ad totam BE. Exprimat AB velocitatem, quam in punctis quiescentibus gigneret 
arcus ille repulsivus per suam aream, una cum differentia omnium sequentium arcuum 
repulsivorum supra omnes sequentes attractivos : exprimat autem BC velocitatem, cum 
qua advenitur ad distantiam respondentem 

initio ejus arcus : exprimet AC velocitatem, 
qu33 habebitur, ubi jam distantia evasit major, 
& vis insensibilis, ac ejus excessus supra 
priorem AB erit BD, exiguus sane etiam re- 
spectu BC, si BC fuerit exigua respectu AB, 
adeoque multo magis respectu AB ; & ob 
eandem rationem perquam exigua area sequentis 
cruris attractivi ingentem illam jam acqui- 
sitam velocitatem nihil ad sensum mutabit, quae 
permanebit ad sensum eadem post recessum in 

197. Haec accident binis punctis sibi relictis, vel impulsis [90] in recta, qua junguntur, 
cum oppositis velocitatibus aequalibus, quo casu etiam facile demonstratur, punctum, 
quod illorum distantiam bifariam secat, debere quiescere ; nunquam in hisce casibus 
poterit motus extingui in adventu ad distantiam limitis cohaesionis, & multo minus poterunt 
ea bina puncta consistere extra distantiam limitis cujuspiam, ubi adhuc habeatur vis aliqua 
vel attractiva, vel repulsiva. Verum si alia externa puncta agant in ilia, poterit res multo 
aliter se habere. Ubi ex. gr. a se recedunt, & velocitates recessus augeri deberent in accessu 
ad distantiam limitis cohaesionis ; potest externa compressio illam velocitatem minuere, 
& extinguere in ipso appulsu ad ejusmodi distantiam. Potest externa compressio cogere 
ilia puncta manere immota etiam in ea distantia, in qua se validissime repellunt, uti duae 
cuspides elastri manu compressae detinentur in ea distantia, a qua sibi relictas statim 
recederent : & simile quid accidere potest vi attractivae per vires externas distrahentes. 

198. Turn vero diligenter notandum discrimen inter casus varies, quos inducit varia 
arcuum curvae natura. Si puncta sint in distantia alicujus limitis cohassionis, circa quern 
sint arcus amplissimi, ita, ut proximi limites plurimum inde distent, & multo magis etiam, 
quam sit tota distantia proximi citerioris limitis ab origine abscissarum ; turn poterunt 
externa vi comprimente, vel distrahente redigi ad distantiam multis vicibus minorem, 
vel majorem priore ita, ut semper adhuc conentur se restituere ad priorem positionem 
recedendo, vel accedendo, quod nimirum semper adhuc sub arcu repulsive permaneat, vel 
attractive. At si ibi frequentissimi limites, curva saepissime secante axem ; turn quidem 
post compressionem, vel distractionem ab externa vi factam, poterunt sisti in multo minore, 
vel majore distantia, & adhuc esse in distantia alterius limitis cohaesionis sine ullo conatu 
ad recuperandum priorem locum. 

Superiorum usus in 199. Haec omnia aliquanto fusius considerare libuit, quia in applicatione ad Physicam 

magno usui erunt infra haec ipsa, & multo magis hisce similia, quae massis respondent 
habentibus utique multo uberiores casus, quam bina tantummodo habeant puncta. Ilia 
ingens agitatio cum oscillationibus variis, & motibus jam acceleratis, jam retardatis, jam 
retro reflexis, fermentationes, & conflagrationes exhibebit : ille egressus ex ingenti arcu 


arc of the last branch which represents gravity. Then indeed the velocity acquired through 
that arc can never be stopped by the subsequent arcs, & the points will recede from one 
another to an immense distance. Nay further, if that repulsive arc taken together with 
the subsequent repulsive arcs has a very great excess of area over the subsequent attractive 
arcs, then the points will continue to recede to an immense distance from one another 
with a very great velocity ; &, although points arrive at this repulsive arc, which is so strong, 
with considerably different velocities, yet the velocities after this fresh & exceedingly great 
increase will be very little different from one another. For, if to the square of a very 
great number there is added the square of a number that is much less, although 
not in itself very small, the square root of the sum differs very little from the first 

196. This indeed is very evident from Euclidean geometry even. In Fig. 20, let The demonstration 
AB be a fairly long line, to which is added, perpendicular to it, BC, which is much less ls P erfectl y sun P Ie - 
than AB. Then, with centre A, & radius AC, describe a semicircle meeting AB on either 

side in E & D. On adding the square on BC to the square on AB, we get the square on 
AC or AD ; & yet this exceeds the former root AB by BD only, which is always less than 
BC, bearing the same ratio to it as BC bears to the whole length BE. Suppose that 
AB represents the velocity which the repulsive arc, owing to the area under it, would 
generate in points initially at rest, together with the difference for all the subsequent 
repulsive arcs over all the subsequent attractive arcs ; also let BC represent the velocity 
with which the distance corresponding to the beginning of this arc is reached ; then AC 
will represent the velocity which is obtained when the distance has already become of 
considerable amount, & the force insensible. Now the excess of this above the former 
velocity AB will be represented by BD ; & this is really very small compared with BC, 
if BC were very small compared with AB ; & therefore much more so with regard to AB. 
For the same reason, the very small area under the subsequent attractive branch will not 
sensibly change the very great velocity acquired so far ; this will remain sensibly the same 
after recession to a huge distance. 

197. These things will take place in the case of two points left to themselves, or impelled What may happen 
along the straight line joining them with velocities that are equal & opposite ; in such the^areT^then? 
a case it can be easily proved that the middle point of the distance between them is bound selves ; what may 
to remain at rest. The motion in the cases we have discussed can never be destroyed ^teif 11 under th the 
altogether on arrival at a distance equal to that of a limit-point of cohesion, & much less actions of other 
will the two points be able to stop at a distance apart that is not equal to that of some them 3 extemal to 
limit-point, as far as which there is some force acting, either attractive or repulsive. But 

if other external points act upon them, we may have altogether different results. For 
instance, in a case where they recede from one another, & the velocities would therefore 
be bound to be increased as they approached a distance equal to that of a limit-point of 
cohesion, an external compression may diminish that velocity, & completely destroy it 
as it approaches the distance of that limit-point. An external compression may even 
force the points to remain motionless at a distance for which they repel one another very 
strongly ; just as the two ends of a spring compressed by the hands are kept at a distance 
from which if left to themselves they will immediately depart. A similar thing may come 
about in the case of an attractive force when there are external tensile forces. 

198. Now, a careful note must be made of the distinctions between the various cases, if the limit-points 
which arise from the various natures of the arcs of the curve. If our points are at a distance fj e far a part, there 

.,_. i i 111 is a tenaency to 

of any limit-point of cohesion, on each side of which the arcs are very wide, so that the return if the dis- 
nearest limit-points are very far distant from it, & also much more so than the nearest j^sfderabSH-han^e; 
limit-point to the left is distant from the origin of abscissae ; they may, under the action but this is not the' 
of an external force causing either compression or tension, be reduced after many alternations l^tsare very'dose 
to a distance, either less, or greater, than the original distance, in such a way that they together, 
will always strive however to revert to their old position by receding from or approaching 
towards one another ; for indeed they will still always remain under a repulsive, or an 
attractive arc. But if, near the limit-point in question, the limit-points on either side 
occur at very frequent intervals ; then indeed, after compression, or separation, caused 
by an external force, they may stop at a much less, or a much greater, distance apart, & 
still be at a distance equal to that of another limit-point of cohesion, without there being 
any endeavour to revert to their original position. 

199. All these considerations I have thought it a good thing to investigate somewhat The use of the 
at length ; for they will be of great service later in the application of the Theory to physics, ^ ve facts "* phy " 
both these considerations, & others like them to an even greater degree ; namely those 

that correspond to masses, for which indeed there are far more cases than for a system 
of only two points. The great agitation, with its various oscillations & motions that are 
sometimes accelerated, sometimes retarded, & sometimes reversed, will represent fermentations 


repulsive cum velocitatibus ingentibus, quas ubi jam ad ingentes deventum est distantias, 
parum admodum a se invicem differant, nee ad sensum mutentur quidquam per immensa 
intervalla, luminis emissionem, & propagationem uniformem., ac ferme eandem celeritatem 
in quovis ejusdem speciei radio fixarum, Solis, flammse, cum exiguo discrimine inter 
diversos coloratos radios ; ilia vis permanens post compressionem ingentem, vel diffractionem 
elasticitati explicandae in-[9i]-serviet ; quies ob frequentiam limitum, sine conatu ad 
priorem recuperandam figuram, mollium corporum ideam suggeret ; quae quidem hie 
innuo in antecessum, ut magis haereant animo, prospicienti jam nine insignes eorum usus. 

Motus binorum 2OO. Quod si ilia duo puncta proiiciantur oblique motibus contrarns, & aequalibus 

punctorum oblique j- -11 j i rr 

projectorum. P er directiones, quae cum recta jungente ipsa ilia duo puncta angulos aequales efficiant ; 

turn vero punctum, in quo recta ilia conjungens secatur bifariam, manebit immotum ; 
ipsa autem duo puncta circa id punctum gyrabunt in curvis lineis aaqualibus, & contrariis, 
quae data lege virium per distantias ab ipso puncto illo immoto (uti daretur, data nostra 
curva virium figurae i, cujus nimirum abscissae exprimunt distantias punctorum a se invicem, 
adeoque eorum dimidiae distantias a puncto illo medio immoto) invenitur solutione pro- 
blematis a Newtono jam olim soluti, quod vocant inversum problema virium centralium, 
cujus problematis generalem solutionem & ego exhibui syntheticam eodem cum Newtoniana 
recidentem, sed non nihil expolitam, in Stayanis Supplementis ad lib. 19. 

Casus, in quo duo 201. Hie illud notabo tantummodo, inter infinita curvarum genera, quae describi 

sc 1 Hbere debe &pira < ies P ossunt 5 cum nulla sit curva, quas assumpto quovis puncto pro centre virium describi 
circa medium im- non possit cum quadam virium lege, quae definitur per Problema directum virium 
centralium, esse innumeras, quas in se redeant, vel in spiras contorqueantur. Hinc fieri 
potest, ut duo puncta delata sibi obviam e remotissimis regionibus, sed non accurate in 
ipsa recta, quae ilia jungit (qui quidem casus accurati occursus in ea recta est infinities 
improbabilior casu deflexionis cujuspiam, cum sit unicus possibilis contra infinites), non 
recedant retro, sed circa punctum spatii medium immotum gyrent perpetuo sibideinceps 
semper proxima, intervallo etiam sub sensus non cadente ; qui quidem casus itidem 
diligenter notandi sunt, cum sint futuri usui, ubi de cohaesione, & mollibus corporibus 
agendum erit. 

Theorema de statu 202. Si utcunque alio modo projiciantur bina puncta velocitatibus quibuscunque ; 

enSfter m 1n U 'maSs P test facile ostendi illud : punctum, quod est medium in recta jungente ipsa, debere 

centri gravitatis quiescere, vel progredi uniformiter in directum, & circa ipsum vel quietum, vel uniformiter 

progrediens, debere haberi vel illas oscillationes, vel illarum curvarum descriptiones. 

Verum id generalius pertinet ad massas quotcunque, & quascunque, quarum commune 

gravitatis centrum vel quiescit, vel progreditur uniformiter in directum a viribus mutuis 

nihil turbatum. Id theorema Newtonus proposuit, sed non satis demonstravit. Demon- 

strationem accuratissimam, ac generalem simul,& non per casuum inductionem tantummodo, 

inveni, ac in dissertation e De Centra Gravitatis proposui, quam ipsam demonstrationem 

hie etiam inferius exhibebo. 

Accessum aiterius [92] 203. Interea hie illud postremo loco adnotabo, quod pertinet ad duorum 

quodvis ad aiterius punctorum motum ibi usui futurum : si duo puncta moveantur viribus mutuis tantummodo, 

aequari recessui ex & ultra ipsa assumatur planum quodcunque ; accessus aiterius ad illud planum secundum 

directionem quamcunque, aequabitur recessui aiterius. Id sponte consequitur ex eo, 

quod eorum absoluti motus sint aequales, & contrarii ; cum inde fiat, ut ad directionem 

aliam quamcunque redacti aequales itidem maneant, & contrarii, ut erant ante. Sed de 

aequilibrio, & motibus duorum punctorum jam satis. 

Transitus ad syste- 204. Deveniendo ad systema trium punctorum, uti etiam pro punctis quotcunque, 

trium "binagene res > si generaliter pertractari deberet, reduceretur ad haac duo problemata, quorum alterum 
alia probiemata. pertinet ad vires, & alterum ad motus : I. Data positions, 5? distantia mutua eorum punc- 
torum, invenire magnitudinem, & directionem vis, qua urgetur quodvis ex ipsis, composites a 
viribus, quibus urgetur a reliquis, quarum singularum virium lex communis datur per curvam 
figure primce. 2. Data ilia lege virium figures -primes invenire motus eorum punctorum, 
quorum singula cum datis velocitatibus projiciantur ex datis locis cum datis directionibus. 
Primum facile solvi potest, & potest etiam ope curva; figurae i determinari lex virium 


& conflagrations. The starting forth from a very large repulsive arc with very great 
velocities, which, as soon as very great distances have been reached, are very little different 
from one another ; nor are they sensibly changed in the slightest degree for very great 
intervals ; this will represent the emission & uniform propagation of light, & the approximately 
equal velocities in any ray of the same kind from the stars, the sun, and a flame, with a 
very slight difference between rays of different colours. The force persisting after 
compression, or separation, will serve to explain elasticity. The lack of motion due to 
the frequent occurrence of limit-points, without any endeavour towards recovering the 
original configuration, will suggest the idea of soft bodies. I mention these matters here 
in anticipation, in order that they may the more readily be assimilated by a mind that 
already sees from what has been said that there is an important use for them. 

200. But if the two points are projected obliquely with velocities that are equal and The motion of two 
opposite to one another, in directions making equal angles with the straight line joining obikmei pro ^ ected 
the two points ; then, the point in which the straight line joining them is bisected 

will remain motionless ; the two points will gyrate about this middle point in equal curved 
paths in opposite directions. Moreover, if the law of forces is given in terms of the distances 
from that motionless point (as it will be given when our curve of forces in Fig. i is given, 
where the abscissae represent the distances of the points from one another, & therefore 
the halves of these abscissae represent -the distances from the motionless middle point), 
then we arrive at a solution of the problem already solved by Newton some time ago, which 
is called the inverse -problem of central forces. Of this problem I also gave a general synthetic 
solution that was practically the same thing as that of Newton, not altogether devoid of 
neatness, in the Supplements to Stay's Philosophy, Book 3, Art. 19. 

201. At present I will only remark that, amongst the infinite number of different The case in which 
curves that can be described, there are an innumerable number which will either re-enter boumT^o^teswibe 
their paths, or wind in spirals ; for there is no curve that, having taken any point whatever spirals about the 
for the centre of forces, cannot be described with some law of forces, which is determined j^^ nless mlddle 
by the direct problem of central forces. Hence it may happen that two points approaching 

one another from a long way off, but not exactly in the straight line joining them and 
the case of accurate approach along the straight line joining them is infinitely more improbable 
than the case in which there is some deviation, since the former is only one possible case 
against an infinite number of others then the points will not reverse their motion and 
recede, but will gyrate about a motionless middle point of space for evermore, always 
remaining very near to one another, the distance between them not being appreciable by 
the senses. These cases must be specially noted ; for they will be of use when we come 
to consider cohesion & soft bodies. 

202. If two points are projected in any manner whatever with any velocities whatever, Theorem on the 
it can readily be proved that the middle point of the line joining them must remain at steady state of the 

1 . , r , . , , . r , , , , J . P , , . . central point &, 

rest or move uniformly in a straight line ; and that about this point, whether it is at rest more generally, 
or is moving uniformly, the oscillations or descriptions of the curved paths, referred to of the . cen , tre of 

i ' T> i 11 i r c gravity in the case 

above, must take place. But this, more generally, is a property relating to masses, of any of masses. 

number or kind, for which the common centre of gravity is either at rest or moves uniformly 

in a straight line, in no wise disturbed by the mutual forces. This theorem was enunciated 

by Newton, but he did not give a satisfactory proof of it. I have discovered a most rigorous 

demonstration, & one that is at the same time general, & I gave it in the dissertation 

De Centra Gravitatis ; this demonstration I will also give here in the articles that 


203. Lastly, I will here mention in passing something that refers to the motion of The approach of 
two points, which will be of use later, in connection with that subject. If two points poLts towards^y 
move subject to their mutual forces only, & any plane is taken beyond them both, then plane is equal to 
the approach of one of them to that plane, measured in any direction, will be equal to the other^rom 1 it 'on 
recession of the other. This follows immediately from the fact that their absolute motions account of 'the 
are equal & opposite ; for, on that account, it comes about that the resolved parts in any mutual force - 
other direction also remain equal & opposite, as they were to start with. However, I 

have said enough for the present about the equilibrium & motions of two points. 

204. When we come to consider systems of three points, as also systems of any number Extension to a 
of points, the whole matter in general will reduce to these two problems, of which the system of three 

tt 11 . T, . . . , points ; two general 

one refers to forces and the other to motions, i. Being given the position and the mutual problems. 

distance of the points, it is required to find the magnitude and direction of the force, to which 

any one of them is subject ; this force being the resultant of the forces due to the remaining 

points, and each of these latter being found by a general law which is given by the curve of Fig. i. 

2. Being given the law of forces represented by Fig. I, it is required to find the motions of 

the points, when each of them is projected with known velocities from given initial positions 

in given directions. The first of these problems is easily solved ; and also, by the aid of 


generaliter pro omnibus distantiis assumptis in quavis recta positionis datae,' a que id tarn 
geometrice determinando per puncta curvas, quae ejusmodi legem exhibeant, ac determinent 
sive magnitudinem vis absolutae, sive magnitudines binarum virium, in quas ea concipiatur 
resoluta, & quarum altera sit perpendicularis data? illi rectse, altera secundum illam agat ; 
quam exhibendo tres formulas analyticas, quas id praestent. Secundum omnino generaliter 
acceptum, & ita, ut ipsas curvas describendas liceat definire in quovis casu vel constructione, 
vel caculo, superat (licet puncta sint tantummodo tria) vires methodorum adhuc cognit- 
arum : & si pro tribus punctis substituantur tres massae punctorum, est illud ipsum 
celeberrimum problema quod appellant trium corporum, usque adeo qusesitum per haec 
nostra tempora, & non nisi pro peculiaribus quibusdam casibus, & cum ingentibus limita- 
tionibus, nee adhuc satis promoto ad accurationem calculo, solutum a paucissimis nostri 
asvi Geometris primi ordinis, uti diximus num. 122. 
Theorema de motu 205. Pro hoc secundo casu illud est notissimum, si tria puncta sint in fig. 21 A, C, B, 

puncti habentis ac- ,. J . . T. i > IT- T-> i 

tionem cum aiiis & distantia AB duorum divisa semper bifanam in D, ac ducta CD, & assumpto ejus 
binis - triente DE, utcunque moveantur eadem puncta 

motibus compositis a projectionibus quibus- 

cunque, & mutatis viribus ; punctum E debere 

vel quiescere semper, vel progredi in directum 

motu uniformi. Pendet id a general! theore- 

mate de centre gravitatis, cujus & superius 

injecta est mentio, & de quo age-[93]-mus 

infra pro massis quibuscunque. Hinc si sibi re- 

linquantur, accedet C ad E, & rectae AB 

punctum medium D ibit ipsi obviam versus 

ipsum cum velocitate dimidia ejus, quam ipsum 

habebit, vel contra recedent, vel hinc, aut inde 

movebuntur in latus, per lineas tamen similes, 

atque ita, ut C, & D semper respectu puncti E 

immoti ex adverse sint, in quo motu tam directio . _ 

rectae AB, quam directio rectae CD, & ejus incli- " _ *^ 

natio ad AB, plerumque mutabitur. 
Determmatio vis 2 o6. Quod pertinet ad inveniendam vim pro quacunque positione puncti C respectu 

ejusdem composite A T> r -i T r 

e binis viribus. punctorum A, & B, ea facile sic mvemetur. In fig. i assumendae essent abscissae in axe 
asquales rectis AC, BC figurae 21, & erigendae ordinatas ipsis respondentes, quae vel ambae 
essent ex parte attractiva, vel ambae ex parte repulsiva ; vel prima attractiva, & secunda 
repulsiva ; vel prima repulsiva & secunda attractiva. In primo casu sumendae essent CL, 
CK ipsis aequales (figura 21 exhibet minores, nenimis excrescat) versus A, &B ; in secundo 
CN, CM ad partes oppositas A,B : in tertio CL versus A, & CM ad partes oppositas B ; in 
quarto CN ad partes oppositas A, & CK versus B. Tam complete parallelogrammo LCKF, 
vel MCNH, vel LCMI, vel KCNG, diameter CF, vel CH, vel CI, vel CG exprimeret 
directionem, & magnitudinem vis compositae, qua urgetur C a reliquis binis punctis. 

Methodus constru- 207. Hinc si assumantur ad arbitrium duo loca qusecunque punctorum A, & B, ad 

expi? <l u3e referendum sit tertium C ; ducta quavis recta DEC indefinita, ex quovis ejus puncto 

mat vim ejusmodi. posset erigi recta ipsi perpendicularis, & asqualis illi diametro, ut CF in primo casu, ac 
haberetur curva exprimens vim absolutam puncti in eo siti, & solicitati a viribus, quas 
habet cum ipsis A, & B. Sed satis esset binas curvas construere, alteram, quae exprimeret 
vim redactam ad directionem DC per perpendiculum FO, ut CO ; alteram, quae exprimeret 
vim perpendicularem OF : nam eo pacto haberentur etiam directiones vis absolutae ab 
iis compositae per ejusmodi binas ordinatas. Oporteret autem ipsam ordinatam curvas 
utriuslibet assumere ex altera plaga ipsius CD, vel ex altera opposita ; prout CO jaceret 
versus D, vel ad plagam oppositam pro prima curva; & prout OF jaceret ad alteram partem 
rectae DC, vel ad oppositam, pro secunda. 

Expressio magis 208. Hoc pacto datis locis A, B pro singulis rectis egressis e puncto medio D duas 

* 115 Per super " haberentur diversae curvae, quae diversas admodum exhiberent virium leges ; ac si quasre- 
retur locus geometricus continuus, qui exprimeret simul omnes ejusmodi leges pertinentes 
ad omnes ejusmodi curvas, sive indefinite exhiberet omnes vires pertinentes ad omnia 


the curve given in Fig. i, the law of forces can be determined in general for any assumed 
distances along any straight line given in position. Moreover, this can be effected either 
by constructing geometrically curves through sets of points, which represent a law of this 
sort & give either the magnitude of the absolute force, or the magnitudes of the pair of 
forces into which it may be considered to be resolved, the one acting perpendicularly to 
the given straight line & the other in its direction ; or else by writing down three analytical 
formulae, which will represent its value. The second, if treated perfectly generally, & 
in such a manner that the curves to be described can be assigned in any case whatever, 
either by construction or by calculation, is (even when there are only three points in question) 
beyond the power of all methods known hitherto. Further, if instead of three points 
we have three masses of points, then we have the well-known problem that is called " the 
problem of three bodies." The solution of this problem is still sought after in our own 
times ; & has only been solved in certain special cases, with great limitations by a very 
few of the geometricians of our age belonging to the highest rank, & even then with insufficient 
accuracy of calculation ; as was pointed out in Art. 122. 

205. As for this second case, it is very well known that, if in Fig. 21, A,C,B, are three Theorem with re- 
points, & the distance between two of them, A & B, is always bisected at D, & CD is joined, g'Vyjj m ^ r 
& DE is taken equal to one third of DC, then, however these points move under the influence the action of two 
of the forces compounded from the forces of any projection whatever & the mutual forces, other P mts - 

the point E must always remain at rest or proceed in a straight line with uniform motion. 
This depends on a general theorem with regard to the centre of gravity, about which 
passing mention has already been made, & with which we shall deal in what follows for the 
case of any masses whatever. From this it follows that, if they are left to themselves, 
the point C will approach the point E, & D, the middle point of the straight line AB, will 
move in the opposite direction towards E with half the velocity of C ; or, on the contrary, 
both C & D will recede from E ; or they will move, one in one direction & the other in 
the opposite direction : nevertheless they will follow similar paths, in such a manner that 
C & D will always be on opposite sides of the stable point E ; & in this motion, the direc- 
tion of the straight line AB, that of the straight line DE, & the inclination of the latter 
to AB will usually be altered. 

206. As regards the determination of the force for any position of the point C with Determination o f 
regard to the points A & B, that is easily effected in the following manner. Take, in Fig. I, compound^ 
abscissa measured along the axis equal to the straight lines AC & BC of Fig. 21 ; draw two forces. 

the ordinates corresponding to them, which may be either both on the attractive side of 
the axis, or both on the repulsive side ; or the first on the attractive & the second on the 
repulsive ; or the first on the repulsive & the second on the attractive side. In the first 
case, take CL, CK, equal to these ordinates (in Fig. 21 they are reduced so as to prevent 
the figure from being too large) ; let them be taken in the direction of A & B ; similarly, 
in the second case, take CN & CM in the opposite directions to those of A & B ; and, in 
the third case, take CL in the direction of A, & CM in the direction opposite to that of B ; 
whilst, in the fourth case, take CN in the direction opposite to that of A, & CK in the 
direction of B. Then, completing the parallelogram LCKF, or MCNH, or LCMI, or 
KCNG, the diagonal CF, or CH, or CI, or CG, will represent the direction & the magnitude 
of the resultant force, which is exerted upon the point C by the remaining two points. 

207. Hence, if any two positions are taken at random as those of the points A & B, The method of 
& to these the third point C is referred ; & if any straight line DEC is drawn of indefinite curve' wWch^m in 
length ; then from any point of it a straight line can be erected perpendicular to it, & general express a 
equal to the diagonal of the parallelogram, for instance CF in the first case. From these force of this sort ' 
perpendiculars a curve will be obtained, which will represent the absolute force on a point 

situated in the straight line DEC, & under the action of the forces exerted upon it by the 
points A & B. However, it would be more satisfactory if two curves were constructed ; 
one of which would represent the force resolved along the direction DC by means of a 
perpendicular FO, such as CO ; & the other to represent the perpendicular force OF. 
For, in this way, we should also obtain the directions of the absolute forces compounded 
from these resolved parts, by means of the two ordinates of this kind. Moreover, 
we ought to take these ordinates of either of the curves on the one side or the other of 
the straight line CD, according as CO would be towards D, or away from it, in the first 
curve, & according as OF would be away from the straight line CD, on the one side or on the 
other, in the second curve. 

208. In this way, given the positions of A & B, for each straight line drawn through A m re g eneral 
the point D, we should obtain distinct curves ; & these would represent altogether different of ^ surface. 71 
laws of forces. If then a continuous geometrical locus is required, which would 
simultaneously represent all the laws of this kind relating to every curve of this sort, 

or express in general all the forces pertaining to all points such as C, wherever they might 

I 5 8 


puncta C, ubicunque collocata ; oporteret erigere in omnibus punctis C rectas normales 
piano ACB, alteram aequalem CO, [94] alteram OF, & vertices ejusmodi normalium 
determinarent binas superficies quasdam continuas, quarum altera exhiberet vires in 
directione CD attractivas ad D, vel repulsivas respectu ipsius, prout, cadente O citra, vel 
ultra C, normalis ilia fuisset erecta supra, vel infra planum ; & altera pariter vires perpen- 
diculares. Ejusmodi locus geometricus, si algebraice tractari deberet, esset ex iis, quos 
Geometrse tractant tribus indeterminatis per unicam aequationem inter se connexis ; ac 
data aequatione ad illam primam curvam figurse I, posset utique inveniri tam sequatio ad 
utramlibet curvam respondentem singulis rectis DC, constans binis tantum indeterminatis, 
quam sequatio determinans utramlibet superficiem simul indefinite per tres indetermin- 
atas. () 

Methpdus determi- [gel 20Q. Si pro duobus punctis tantummodo agentibus in tertium daretur numerus 

nandi vim composi- L7 T J . r . , . . . . ., 

tam ex viribus re- quicunque punctorum positorum in datis locis, ac agentium in idem punctum, posset utique 
spicientibus puncta constructione simili inveniri vis, qua sineula agunt in ipsum collocatum in quovis assumpto 

quotcunque. , . . ' ~*. . .. . r , ., . .? 

loci puncto, ac vis ex ejusmodi viribus composita denniretur tam directione, quam 
magnitudine, per notam virium compositionem. Posset etiam analysis adhiberi ad expri- 
mendas curvas per asquationes duarum indeterminatarum pro rectis quibuscunque, & (") 
si omnia puncta jaceant in eodem piano, superficies per asquationem trium. [96] Mirum 
autem, quanta inde diversarum legum combinatio oriretur. Sed & ubi duo tantummodo 
puncta agant in tertium, incredibile dictu est, quanta diversitas legum, & curvarum inde 
erumpat. Manente etiam distantia AB, leges pertinentes ad diversas inclinationes rectae 
DC ad AB, admodum diversse obveniunt inter se : mutata vero punctorum A, B distantia 

(n) Stantibus in fig. 22 punctis ADBCKFLO, ut in fig. 21, ducantur perpendicula BP, AQ in CD, qute dabuntur 
data inclinations DC, y punctis B, A, ac pariter dabuntur y DP, DQ. Dicatur prtsterea DC = x, y dabuntur analytice 
CQ, CP. Quare ob angulos rectos P, Q, dabuntur etiam analytice CB, CA. Denominentur CK=, CL =z, CF =y. 
Quoniam datur AB, y dantur analytice AC, CB ; dabitur analytice ex applicatione Algebrte ad Trigonometriam 
sinus anguli ACB per x, y datas quantitates, qui est idem, ac sinus anguli CKF complements ad duos rectos. Datur 
autem idem ex datis analytice valoribus CK = , KF = CL =z, CF =y ; quare habetur ibi una tequatio per x, y, 
z, u, y constantes. Si pr<eterea valor CB ponatur pro valore abscissae in tequatione curvte figurte I ; acquiritur altera 
tequatio per valores CK, CB, sive per x,u, y constantes. Eodem facto invenietur ope tequationis curvte figure I tertia 
tequatio per AC, & CL, adeoque per x, z, y constantes. Quare jam babebuntur tequationes tres per x,u,z,y, y con- 
stantes, qute, eliminates u, y z, reducentur ad unicam per x,y, & constantes, ac ea primam illam curvam definiet. 

Quod si queeratur' tequatio ad secundam curvam, cujus ordinata est CO, vel tertiam, cujus ordinata OF, 


inveniri itidem poterit. Nam datur analytice sinus anguli DCB = , W * trianguk FCK datur analytice 


sinus FCK ===- X sin CKF. Quare datur analytice etiam sinus 

differentite OCF, adeoque & ejus cosinus, & inde, ac ex CF datur 
analytice OF, vel CO. Sz igitur altera ex illis dicatur p, acquiri- 
tur nova tequatio, cujus ope una cum superioribus eliminari 
poterit pristerea una alia indeterminata ; adeoque eliminata 
CF =y, habebitur unica tequatio per x,p, y constantes, qua 
exhibebit utramlibet e reliquis curvis determinantibus legem 
virium CO, vel OF. 

Pro tequatione cum binis indeterminatis, quts exhibebit locum- 
ad superficiem, ducatur CR perpendicularis ad AB, y dicatur 
DR x, RC = q, denominatis, ut prius, CK =, CL = z, 
CF = v ; y quoniam dantur AD, DB ; dabuntur analytice per x, 
y constantes AR, RB, adeoque per x, q, & constantes AC, CB, W 
factis omnibus reliquis, ut prius, kabebuntur quatuor tequationes 
per x,q,u,z,y,p, y constantes, qute eliminatis valoribus u,z,y, 
reducentur ad unicam datam per constantes, y tres indeterminatas 
x,p,q, sive DR, RC, y CO, vel OF, qute exhibebit qutesitum 
locum ad superficiem. 

Calculus quidem esset immensus, sed patet methodus, qua deveniri possit ad tequationem qutesitam. Mirum autem, 
quanta curvarum, y superficierum, adeoque y legum virium varietas obvenerit, mutata tantummodo distantia AB binorum 
punctorum agentium in tertium, qua mutata, mutatur tola lex, y tequatio. 

(o) Htec conditio punctorum jacentium in eodem piano necessaria fuit pro loco ad superficiem, y pro tequatione, qute 
legem virium exhibeat per tequationem indeterminatarum tantummodo trium : at si puncta sint plura, y in eodem piano 
non jaceant, quod punctis tantummodo tribus accidere omnino non potest ; turn vero locus ad superficiem, y tequatio trium 
indeterminatarum non sufficit, sed ad earn generaliter exprimendam legem Geometria omnis est incapax, y analysis indiget 
tequatione indeterminatarum quatuor. Primum patet ex eo, quod si manentibus punctis A, B, exeat punctum C ex data 
quodam piano, pro quo constructus sit locus ad superficiem ; liceret converters circa rectam AB planum illud cum superficie 
curva legem virium determinate, donee ad punctum C deveniret planum ipsum : turn enim erecto perpendiculo usque ad 
superficiem illam curvam, definiretur per ipsum vis agens secundum rectam CD, vel ipsi perpendicularis, prout locus ille 
ad curvam superficiem constructus fuerit pro altera ex iis. 


be situated ; we should have to erect at every point C normals to the plane ACB, one of 
them equal to CO & the other to OF. The ends of these normals would determine two 
continuous surfaces ; & of these, the one would represent the forces in the direction CD, 
attractive or repulsive with respect to the point D, according as the normal was erected 
above or below this plane, whether C fell on the near side or on the far side of D ; & 
similarly the other would represent the perpendicular forces. A geometrical locus of this 
kind, if it has to be treated algebraically, is such as geometricians deal with by means of 
three unknowns connected together by a single equation ; &, if the equation to the primary 
curve of Fig. i is given, it would in all cases be possible to find, not only the equations to 
the two curves corresponding to each & every straight line DC, involving only two unknowns, 
but also the equations for both the surfaces corresponding to the general determination, 
by means of three unknowns. () 

209. If instead of only two points acting upon a third we are given any number of The method of 
points situated in given positions, & acting on the same point, it would be possible, by a force ""compounded 
similar construction in each case, to find the force, with which each acts on the point from the forces due 
situated in any chosen position ; & the force compounded from forces of this kind would points 7 The 1 great 
be determined, both in position & magnitude, by the well-known method for composition "]^ s r & variet y 
of forces. Also analysis could be employed to represent the curves by equations involving 
two unknowns for any straight lines ; & () provided that all the points were in the same 
plane, the surface could be represented by an equation involving three unknowns. But 
it is marvellous what a huge number of different laws arise. But, indeed, it is incredible, 
even when there are only two points acting on a third, how great a number of different 
laws & curves are produced in this way. Even if the distance AB remains the same, the 
laws with respect to different inclinations of the straight line CD to the straight line AB, 
come out quite different to one another. But when the distance of the points A & B from 

(n) In Fig. 22, let the -points A,D,B,C,K,F,L,O be in the same positions as in Fig. 21, y let BP, AQ be drawn 
perpendicular to CD ; then these will be known, if the inclination of CD y the positions of A y B are known: y 
so also will DP W DQ be known. Further, suppose DC = x, then CQ y CP will be given analytically. Hence on 
account of the right angles at P y Q, CB y CA will also be given analytically. Suppose CK = w, CL = z, CF = y. 
Since AB is known, y AC, CB are given analytically, by an application of algebra to trigonometry, the sine of the 
angle ACB is also known analytically in terms of x y known quantities ; y this is the same thing as the sine of 
the supplementary angle CKF. Moreover the same thing will be given in terms of the known analytical values of 
3K = u, KF = CL = z, CF = y. Hence there is obtained in this case an equation involving x,y,z,u, y constants. 
If, in addition, the value CB is substituted for the value of the abscissa in the equation of the curve in Fig. I, another 
equation will be obtained in terms of the values of CK, CB, i.e. in terms of x, u, y constants. In a similar way by the help 
of the equation of the curve of Fig. I, there can be found a third equation in terms of AC y CL, i.e., in terms of 
#,z, y constants. Now, snce there will be' thus obtained three equations in terms of x,y,z,u, y constants, these, on 
eliminating u,z, will reduce to a single equation involving x,y, y constants ; y this will be the equation defining 
the first curve. 

Again, / the equation to the second curve is required, of which the ordinate is CO, or of a third curve for which 
the ordinate is CF, it will be possible to find either of these as well. For the sine of the angle DCB is analytically 
given, being equal to BP/CB ; y from the triangle FCK, the sine of the angle FCK is given, being equal to 
'CKF.(FK/CF). There fore the sine of the difference OOP is also given analytically, y therefore also its cosine; y 
from this y the value of CF, the value of OF or CO will be given analytically. If then one or the other of them is 
denoted by p, a new equation will be obtained: by the help of this y one of the equations given above, another of the 
unknowns can be eliminated. If then, we eliminate CF = y, a single equation will be obtained in terms of x,p, y 
constants, which will be that of one or other of the remaining curves determining the law of forces for CO or OF. 

For an equation in three unknowns, which will represent the surface, draw CR perpendicular to AB, y let DR=# 
RC = q ; y, as before, let CK = it, CL = z, CF = y. Then, since AD, DB are given, AR y RB are also given 
analytically in terms of x y constants : y therefore AC y CB are given in terms of x,q, y constants : y if all 
the rest of the work is done as before, four equations will be obtained in terms of x,q,u,z,y,p, y constants. These, on 
eliminating the values u,z,y, will reduce to a single equation in terms of constants y the three unknowns x,p,q, or DR, 
RC, y CO or OF ; this equation will represent the surface required. 

The calculation would indeed be enormous ; but the method, by which the required equation might be obtained is 
perfectly clear. But it is wonderful what a great number of curves y surfaces, y therefore of laws of force, would be 
met with, if merely the distance between A y B, the two points which act upon the third, is changed ; for if this 
alone is changed, the whole law is altered y so too is the equation. 

(o) This condition, that the points should all lie in the same plane, is necessary for the determination of the surface, 
y for the equation, which will express the law of the forces by an equation involving only three unknowns. If the points 
are numerous, y they do not all lie in the same plane (which is quite impossible in the case of only three points), then 
indeed a surface locus, y an equation in three unknowns, will not be sufficient; indeed, to express the law generally, 
the whole of geometry is powerless, y analysis requires an equation in four unknowns. The first point is clear from 
the fact that if, whilst the points A y B remain where they were, the point C moves out of the given plane, with 
regard to which the construction for the surface locus was made, it would be right to rotate about the straight line AB 
that plane together with its curved surface, which determines the law of forces, until the plane passes through the point 
C. For then, if a perpendicular is drawn to meet the curved surface, this would define the force acting along the 
straight line CD, or perpendicular to it, according as the locus to the curved surface had been constructed for the one 
or for the other of them. 



a se invicem, leges etiam pertinentes ad eandem inclinationem DC differunt inter se 
plurimum ; & infinitum esset singula persequi ; quanquam earum variationum cognitio, 
si obtineri utcunque posset, mirum in modum vires imaginationis extenderet, & objiceret 
discrimina quamplurima scitu dignissima, & maximo futura usui, atque incredibilem 
Theoriae foecunditatem ostenderet. 

distantiis 1 2IO> ^8 ^ c s i m pliciora quaedam, ac faciliora, & usum habitura in sequentibus, ac in 

ac ejus usus pro applicatione ad Physicam inprimis attingam tantummodo ; sed interea quod ad generalem 
nu'iiaVinUs s^mma P ert i net determinationem expositam, duo adnotanda proponam. Primo quidem in ipsa 
virium simpiicium. trium punctorum combinatione occurrit jam hie nobis praeter vim determinantem ad 
accessum, & recessum, vis urgens in latus, ut in fig. 21, praeter vim CF, vel CH, vis CI, 
vel CG. Id erit infra magno usui ad explicanda solidorum phaenomena, in quibus, 
inclinato fundo virgse solidae, tola virga, & ejus vertex moventur in latus, ut certam ad 
basim positionem acquirant. Deinde vero illud : haec omnia curvarum, & legum discrimina 
tam quae [97] pertinent ad diversas directiones rectarum DC, data distantia punctorum 
A, B, quam quae pertinent ad diversas distantias ipsorum punctorum A, B, data etiam 
directione DC, ac hasce vires in latus haberi debere in exiguis illis distantiis, in quibus 
curva figurae I circa axem contorquetur, ubi nimirum mutata parum admodum distantia, 
vires singulorem punctorum mutantur plurimum, & e repulsivis etiam abeunt in attractivas, 
ac vice versa, & ubi respectu alterius puncti haberi possit attractio, respectu alterius repulsio, 
quod utique requiritur, ut vis dirigatur extra angulum ACB, & extra ipsi ad verticem 
oppositum. At in majoribus distantiis, in quibus jam habetur illud postremum crus 
figurae I exprimens arcum attractivum ad sensum in ratione reciproca duplicata distantiarum, 
vis in punctum C a punctis A, B inter se proximis, utcunque ejusmodi distantia mutetur, 
& quaecunque fuerit inclinatio CD ad AB, erit semper ad sensum eadem, directa ad sensum 
ad punctum D, ad sensum proportionalis reciproce quadrato distantiae DC ab ipso puncto 
D, & ad sensum dupla ejus, quam in curva figurae i requireret distantia DC. 

At secundum sit manifestum ex eo, quod si puncta agenda sint etiam omnia in eodem piano, y punctum, cufus vis 
composita quteritur, in quavis recta posita extra ipsum planum, relationes omnes distantiarum a reliquis punctis, ac 
directionum, a quibus pendent vires singulorum, y compositio ipsarum virium, longe alia essent, ac in quavis recta in eodem 
piano posita, uti facile videre est. Hinc pro quovis puncto loci ubicunque assumpto sua responderet vis composita, y quarta 
aliqua plaga, seu dimensio, prater longum, latum, & profundum, requireretur ad ducendas ex omnibus punctis spatii rectas 
Us viribus proportionales, quarum rectarum vertices locum continuum aliqucm exhiberent determinantem virium legem. 

Sed quod Geometria non assequitur, assequeretur quarta alia dimensio mente concepta, ut si conciperetur spatium totum 
plenum materia continua, quod in mea sententia cogitatione tantummodo effingi potest, W ea esset in omnibus spatii punctis 
densitatis diverse, vel diversi pretii ; turn ilia diversa densitas, vel illud pretium, vel quidpiam ejusmodi, exhibere posset 
legem virium ipsi respondentium, ques nimirum ipsi essent proportionales. Sed ibi iterum ad determmandam directtonem 
vis composite non esset satis resolutio in duas vires, alteram secundum rectam transcuntem per datum punctum ; altcram 
ipsi perpendicularem ; ed requirerentur tres, nimirum vel omnes secundum tres datas directiones, vel tendentes per rectas, 
qua per data tria puncta transeant, vel quavis alia certa lege definitas : adeoque tria loca ejusmodi ad spatium, quarta 
aliqua dimensione, vel qualitate affectum requirerentur, qu<e tribus ejusmodi plusquam Geometricis legibus vis composite 
legem definirent, turn quod pertinet ad ejus magnitudinem, turn quod ad directionem. 

quod non assequitur Geometria, assequeretur Analysis ope aquationis quatuor indeterminatarum : si enim 
planum, quod libuerit, ut ACB, y in eo quavis recta AB, ac in ipsa recta quodvis punitum D ; turn quovis 


conciperetur . M 

hujus segmento DR appellate x, quavis recta RC ipsi perpendiculari y, quavis tertia perpendicular! ad totum planum z, 
per hasce tres indeterminatas involveretur positio puncti spatii cujuscumque, in quo collocatum esset punctum materiel, 
cufus vis quteritur. 

Punctorum agentium utcunque collocatorum ubicunque vel intra id planum, vel extra, possent definiri positiones per 
ejusmodi tres rectas, datas utique pro singulis, si eorum positiones dentur. Per eas, y per illas x,y,z, posset utique haberi 
distantia cujuscumque ex Us punctis agentibus, y positione datis, a puncto indefinite accepto ; adeoque ope aquationis 
figurtz I posset haberi analytice per aquationes quasdam, ut supra, vis ad singula agentia puncta pertinens, y per easdem 
rectas ejus etiam directio resoluta in tres parallelas illis x,y,z. Hinc haberetur analytice omnium summa pro singulis 
ejusmodi directionibus per aliam aquationem derivatam ab ejus summa denominatione, ea nimirum facia = u, ac expunctis 
omnibus subsidiariis valoribus, methodo non absimili ei, quam adhibuimus superius pro loco ad superficiem, deveniretur ad 
unam aquationem constitutam illis quatuor indeterminatis x,y,z,u, y constantibus ; ac tres ejusmodi aquationes pro tribus 
directionibus vim omnem compositam definirent. Sed hac innuisse sit satis, qua nimirum y altiora sunt, y ob ingentem 
complicationcm casuum, ac nostra humantf mentis imbecillitatem nulli nobis inferius futura sunt usui. 


one another is also changed, the laws corresponding to the same inclination of DC are 
altogether different to one another ; & it would be an interminable task to consider them 
all, case by case. However, a comprehensive insight into their variations, if it could be 
obtained, would enlarge the powers of imagination to a marvellous extent ; it would bring 
to the notice a very large number of characteristics that would be well worth knowing & 
most useful for further work ; & it would give a demonstration of the marvellous fertility 
of my Theory. 

210. First of all, therefore, I will here only deal slightly with certain of the more simple The lateral force at 
cases, such as will be of use in what follows, & later when considering the application to tances, Tits use \n 
Physics. But meanwhile, I will enunciate two theorems, applying to the general deter- t] } e consideration 
mination set forth above, which should be noted. Firstly, in the case of the combination absence^ this 

of three points, we have here already met with, in addition to a force inducing approach f ? rce at great 
. . ' . T-,. . IT. ,. X-,T-< ,"ITT ^-.T 2Z^~. rr distances, the sum 

& recession, i.e., in rig. 21, in addition to a force CF or CH, a force CI or CG, urging of the simple forces 

the point C to one side. This will be of great service to us in explaining certain phenomena in the latter case - 

of solids ; for instance, the fact that, if the bottom of a solid rod is inclined, the whole 

rod, including its top, is moved to one side & takes up a definite position with respect to 

the base. Secondly, there is the fact that we are bound to have all these differences of 

curves & laws, not only those corresponding to different directions of the straight lines DC 

when the distance between the points A & B is given, but also those corresponding to 

different distances of the points A & B when the direction of DC is given ; & that we are 

bound to have these lateral forces for very small distances, for which the curve in Fig. I 

twists about the axis ; for then indeed, if the change in distance is very slight, the change 

in the forces corresponding to the several points is very great, & even passes from repulsion 

to attraction & vice versa ; & also there may be attraction for one point & repulsion for 

another ; & this must be the case if the direction of the force has to be without the angle 

ACB, or the angle vertically opposite to it. But, at distances that are fairly large, for 

which we have already seen that there is a final branch of the curve of Fig. i that represents 

attraction approximately in the ratio of the inverse square of the distance, the force on the 

point C, due to two points A & B very near to one another, will be approximately the 

same, no matter how this distance may be altered, or what the inclination of CD to AB 

may be ; its direction is approximately towards D ; & its magnitude will be approximately 

in inverse proportion to the square of DC, its distance from the point D ; that is to say, it 

will be approximately double of that to which in Fig. I the distance DC would correspond. 

The second point is evident from the fact that, if all the points acting are all in the same plane, 5? the point for 
which the resultant farce is required, lies in any straight line situated without that plane, even then all the relations 
between the distances from the remaining points as well as between their directions, will be altogether different from 
those for any straight line situated in the same plane, as can be easily seen. Hence, for any point of space chosen at 
random there would be a corresponding force ; W a fourth region, or dimension, in addition to length, breadth, & depth, 
would be required, in order to draw through each point of space straight lines proportional to these forces, the ends of 
which straight lines would give a continuous locus determining the law for the forces. 

But 'what can not be attained by the use of geometry, could be attained, by imagining another, a fourth, dimension 
(just as if the whole of space were imagined to be full of eontinuous matter, which in my opinion can only be a mental 
fiction) ; W this would be of different density, or different value, at all points of space. Then the different density, or 
value, or something of that kind, might represent the law of forces corresponding to it, these indeed being proportional 
to it. But here again, in order to find the direction of the resultant force, resolution into two forces, the one along the 
straight line passing through the given -point, y the other perpendicular to it, would, not be sufficient. Three resolved 
parts would be required, either all in three given directions, or along straight lines passing through three given points, 
or defined by some other fixed law. Thus, three regions of this kind in space possessed of some fourth dimension or quality 
would be required ; y these would define, by three ultra-geometrical laws of this sort, the law of the resultant force 
both as regards magnitude & direction. 

But what cannot be obtained with the help of geometry could be obtained by the aid of analysis by employing an 
equation with four unknowns. For, if we take any arbitrary plane, as ACB, y in it any straight line AB, y in this 
straight line any point D / then, calling any segment of it x, any straight line perpendicular to it y, y any third 
straight line perpendicular to the whole plane z, there would be contained in these three unknowns the position of any 
point in space, at which is situated a point of matter, for which the force is required. 

The positions of the acting points, however W wherever they may be situated, either within the plane or without 
it, could be defined by three straight lines of this sort ; y these would in all cases be known for each point, if the positions 
of the points are given. By means of these, y the former straight lines denoted by x,y,z, there could he obtained in 
all cases the distance of each of the acting points, that are given in position, from any point assumed indefinitely. Thus 
by the help of the equation to the curve of Fig. I, there could be obtained analytically, by means of certain equations 
similar to those above, the force corresponding to each of the acting points ; also from the same straight lines, its 
direction as well, by resolving along three parallels to x, y, y z. Hence there could be obtained analytically the sum 
of all of them for each of these directions, by means of another equation derived from the symbol used for the sum (for 
instance, let this be called u] ; y, eliminating all the subsidiary values, by a method not unlike that which was used 
above for the surface locus, we should arrive at a single equation in terms of the four unknowns, x, y, z, u, y constants. 
Three equations of this sort, one for each of the three directions, would determine the resultant force completely. But 
let it suffice merely to have mentioned these things ; for indeed they are too abstruse, y, on account of the enormous 
Complexity of cares, y the disability of the human intelligence, will not be of any use to us later. 

1 62 


Demonstratio post- 

remi theorematis. 

2 u. Id quidem facile demonstratur. Si enim AB respectu DC sit perquam exigua, 

, . ,-, ^ . . /-.T^ i i r i A n 

anguius AL.D erit perquam exiguus, & a recta CL) ad sensum bitanam sectus : distantias AC, 
CB erunt ad se invicem ad sensum in ratione sequalitatis, adeoque & vires CL, CK ambae 
attractive debebunt ad sensum aequales esse inter se, & proinde LCKF ad sensum rhombus, 
diametro CF ad sensum secante angulum LCK bifariam, quae rhombi proprietas est, & 
ipsa CF congruente cum CO, ac (ob angulum FCK insensibilem, & CKF ad sensum 
aequalem duobus rectis) aequali ad sensum binis CK, KF, sive CK, CL, simul sumptis ; 
quae singulae cum sint quam proxime in ratione reciproca duplicata distantiarum CB, 
BA ; erunt & eadem, & earum summa ad sensum in ratione reciproca duplicata distantiae 




ingens 212. Porro id quidem commune est etiam massulis constantibus quocunque punctorum 

qu;is mas- _ _ -^ _ ... *** 

exercet in numero. Mutata alarum combmatione, vis composita a vinbus singulorum agens in 
oM- p unc tum distans a massula ipsa per intervallum perquam exiguum, nimirum ejusmodi, 
in remo- in quo curva figurae I circa axem contorquetur, debet mutare plurimum tarn intensitatem 
^us^ quse suanij q uam directionem, & fieri utique potest, quod infra etiam in aliquo simpliciore 
& reciproce, casu trium punctorum videbimus, ut in alia combinatione punctorum massulae pro eadem 
dlS distantia a medio repulsiones praevaleant, in alia attractiones, in alia oriatur vis in latus ad 
perpendiculum, ac in eadem constitutione massulae pro diversis directionibus admodum 
diversae sint vires pro eadem etiam distantia a medio. At in magnis illis distantiis, in 
quibus singulorum punctorum vires jam attractive sunt omnes, & directiones, ob molem 
massulae tarn exiguam respectu ingentis distantias, ad sensum conspirant, vis com-[98] 
-posita ex omnibus dirigetur necessario ad punctum aliquod intra massulam situm, adeoque 
ad sensum ejus directio erit eadem, ac directio rectae tendentis ad mediam massulam, & 
aequabitur vis ipsa ad sensum summae virium omnium punctorum constituentium ipsam 
massulam, adeoque erit attractiva semper, & ad sensum proportionalis in diversis etiam 
massulis numero punctorum directe, & quadrate distantias a medio massulae ipsius reciproce ; 
sive generaliter erit in ratione composita ex directa simplici massarum, & reciproca duplicata 
distantiarum. Multo autem majus erit discrimen in exiguis illis distantiis, si non unicum 
punctum a massula ilia solicitetur, sed massula alia, cujus vis componatur e singulis viribus 
singulorum suorum punctorum, quod tamen in massula etiam respectu massulae admodum 
remotae evanescet, singulis ejus punctis vires habentibus ad sensum aequales, & agentes 
in eadem ad sensum directione ; unde net, ut vis motrix ejus massulae solicitatae, orta ab 
actionibus illius alterius remotae massulae, sit ad sensum proportionalis numero punctorum, 
quas habet ipsa, numero eorum, quae habet altera, & quadrate distantiae, quaecunque sit 
diversa dispositio punctorum in utralibet, quicunque numerus. 

Unde necessaria 213. Mirum sane, quantum in applicatione ad Physicam haec animadversio habitura 

unTf^mitTsTn s ^ usum '> nam inde constabit, cur omnia corporum genera gravitatem acceleratricem 
gravitate, differ- habeant proportionalem massae, in quam tendunt, & quadrato distantiae, adeoque in 
a- superficie Terrae aurum, & pluma cum aequali celeritate descendant seclusa resistentia, vim 
autem totam, quam etiam pondus appellamus, proportionalem praeterea massae suae, adeoque 
in ordine ad gravitatem nullum sit discrimen, quascunque differentia habeatur inter corpora, 
quae gravitant, & in quae gravitant, sed ad solam demum massam, & distantiam res omnis 
deveniat ; at in iis proprietatibus, quae pendent a minimis distantiis, in quibus nimirum 
fiunt reflexionis lucis, & refractiones cum separatione colorum pro visu, vellicationes fibrarum 
palati pro gustu, incursus odoriferarum particularum pro odoratu, tremor communicatus 
particulis aeris proximis, & propagatus usque ad tympanum auriculare pro auditu, asperitas, 
ac aliae sensibiles ejusmodi qualitates pro tactu, tot cohaesionum tarn diversa genera, 
secretiones, nutritionesque, fermentationes, conflagrationes, displosiones, dissolutiones. 
prascipitationes, ac alii effectus Chemici omnes, & mille alia ejusmodi, quae diversa corpora 
a se invicem discernunt, in iis, inquam, tantum sit discrimen, & vires tarn variae, ac tarn 


211. The latter theorem can be easily demonstrated. For, if AB is very small compared S[ oof of thelattcr 
with DC, the angle ACB will be very small, & will be very nearly bisected by the straight 

line CD. The distances AC, CB will be approximately equal to one another ; & thus 
the forces CL, CK, which are both attractive, must be approximately equal to one another. 
Hence, LCKF is approximately a rhombus, & the diagonal CF very nearly bisects the 
angle LCK, that being a property of a rhombus ; CF will fall along CO, &, because the 
angle FCK is exceedingly small & CKF very nearly two right angles, CF will be very 
nearly equal to CK & KF, or CK & CL, taken together. Now each of these are as 
nearly as possible in the inverse ratio of the square of the distances CB, CA ; & these will 
be the same, & their sum therefore approximately inversely proportional to the square 
of the distance DC. 

212. Further this theorem is also true in general for little masses consisting of points, J.*i ere is . a hu f e 

,. T-,, , jjf it- difference in the 

whatever their number may be. Ine force compounded from the several forces acting forces which a small 
on a point, whose distance from the mass is very small, i.e., such a distance as that for which, mass , exerts on , a 

r .-11 7 11 i T i i small mass that 

in Fig. i, the curve is twisted about the axis, must be altered very greatly if the combination is very near to it; 

of the points is altered ; & this is so, both as regards its intensity, & as regards its direction. possibie^nrformH* 

It may even happen, as will be seen later in the more simple case of three points, that in in the forces due 

one combination of the points forming the little mass, & for one & the same distance from these'var 6 dlrectf' 

the mean point, repulsions will preponderate, in another case attractions, & in another as the masses, & 

case there will arise a perpendicular lateral force. Also for the same constitution of the squaref'of the dls 6 

mass, for the same distance from the mean point, there may be altogether different forces tances. 

for different directions. But, for considerable distances, where the forces due to the several 

points are now attractive, & their directions practically coincide owing to the dimensions 

of the little mass being so small compared with the greatness of the distance, the force 

compounded from all of them will necessarily be directed towards some point within the 

mass itself ; & thus its direction will be approximately the same as the straight line drawn 

through the mean centre of the mass ; & the force itself will be equal approximately to 

the sum of all the forces due to the points composing the little mass. Hence, it will always 

be an attractive force ; & in different masses, it will be approximately proportional to the 

number of points directly, & to the square of the distance from the mean centre of the mass 

inversely. That is, in general, it will be in the ratio compounded of the simple direct 

ratio of the masses & the inverse duplicate ratio of the distances. Further, the differences 

will be far greater, in the case of very small distances, if not a single point alone, but 

another mass, is under the action of the little mass under consideration ; for in this case, 

the force is compounded from the several forces on each of the points that constitute it ; 

& yet these differences will also disappear in the case of a mass acted on by a mass considerably 

remote from it, since each of the points composing it is under the influence of forces that 

are approximately equal & act in practically the same direction. Hence it comes about that 

the motive force of the mass acted upon, which is produced by the action of the other 

mass remote from it, is approximately proportional to the number of points in itself, to 

the number of points in the other mass, & to the square of the distance between them, 

whatever the difference in the disposition of the points, or their number, may be for either 


213. It is indeed wonderful what great use can be made of this consideration in the He ice we have 
application of my Theory to Physics ; for, from it it will be clear why all classes of bodies bodies? uniformity 
have an accelerating gravity, proportional to the mass on which they act, & to the square m ^ e c * se ? f 
of the distance [inversely] ; & hence that, on the surface of the Earth, a piece of gold & a uniformity in the 
feather will descend with equal velocity, when the resistance of the air is eliminated. It ca f e of numerous 
will be clear also that the whole force, which we call the weight, is in addition proportional 

to the mass itself ; & thus, without exception, there is no difference as regards gravity, 
no matter what difference there may be between the bodies which gravitate, or towards 
which they gravitate ; the whole matter reducing finally to a consideration of mass & 
distance alone. However, for those properties that depend on very small distances, for 
instance, where we have reflection of light, & refraction with separation of colours, with 
regard to sight, the titillation of the nerves of the palate, with regard to taste, the inrush 
of odoriferous particles where smell is concerned, the quivering motion communicated to 
the nearest particles of the air & propagated onwards till it reaches the drum of the ear 
for sound, roughness & other such qualities as may be felt in the case of touch, the large 
number of kinds of cohesion that are so different from one another, secretion, nutrition, 
fermentation, conflagration, explosion, solution, precipitation, & all the rest of the 
effects met with in Chemistry, & a thousand other things of the same sort, which 
distinguish different bodies from one another ; for these, I say, the differences become 
as great, the forces and the motions become as different, as the differences in the phenomena, 

1 64 


vis in duo puncta 

puacti positi in 

recta jungente 
ipsa. vei in recta 

secante hanc bi- 

fariam, & ad angu- 
los rectos directa 

secundum eandem 


varii motus, qui tarn varia phaenomena, & omnes specificas tot corporum differentias 
inducunt, consensu Theoriae hujus cum omni Natura sane admirabili. Sed hsec, quas 
hue usque dicta sunt ad massas pertinent, & ad amplicationem ad Physicam : interea 
peculiaria quaedam persequar ex innumeris iis, quas per-[99]-tinent ad diversas leges binorum 
punctorum agentium in tertium. 

214. Si libeat considerare illas leges, quas oriuntur in recta perpendiculari ad AB 

, T _ . . , _. . . . T- . . ...... r r . , ... , ,. 

ducta per D, vel m ipsa AB hmc, & inde producta, mprimis facile est videre mud, direc- 
tionem vis compositas utrobique fore eandem cum ipsa recta sine ulla vi in latus, & sine ulla 

. . . " , . . . _ , *, . . _ 

declinatione a recta, quas tendit ad ipsum D, vel ab ipso. Pro recta AB res constat per 
sese . nam v j res Qjgg q U33 a d bina ea puncta pertinent, vel habebunt directionem eandem, 

, t * . . r ,. -1 j 

vel oppositas, jacente ipso tertio puncto in directum cum utroque e pnonbus : unde 
fit, ut vis composita asquetur summae, vel differentias virium singularum componentium, 
quae in eadem recta remaneat. Pro recta perpendiculari facile admodum demonstratur. 
Si enim in fig. 23 recta DC fuerit perpendicularis ad AB sectam bifariam in D, erunt AC, 
BC aequales inter se. Quare vires, quibus C agitatur ab A, & B, sequales erunt, & proinde 
vel ambae attractivae, ut CL, CK, vel ambae repulsivae, ut CN, CM. Quare vis composita 
CF, vel CH, erit diameter rhombi, adeoque secabit bifariam angulum LCK, vel NCM ; 
quos angulos cum bifariam secet etiam recta DC, ob asqualitatem triangulorum DCA, DCB, 
patet, ipsas CF, CH debere cum eadem congruere. Quamobrem in hisce casibus evane- 
scit vis ilia perpendicularis FO, quae in prsecedentibus binis figuris habebatur, ac in iis 
per unicam aequationem res omnis absolvitur (f), quarum ea, quae ad posteriorem casum 
pertinet, admodum facile invenitur. 

casus posterioris. 

2I 5' ^egem pro recta perpendiculari rectae jungenti duo puncta, & asque distanti ab 
utroque exhibet fig. 24, quse vitandae confusionis causa exhibetur, ubi sub numero 24 
habetur littera B, sed quod ad ejus constructionem pertinet, habetur separatim, ubi sub 
num. 24 habetur littera A ; ex quibus binis figuris fit unica ; si puncta XYEAE' censeantur 
utrobique eadem. In ea X, Y sunt duo materiae puncta, & ipsam XY recta CC* secat 
bifariam in A. Curva, quae vires compositas ibi exhibet per ordinatas, constructa est ex 
fig. I, quod fieri potest, inveniendo vires singulas singulorum punctorum, turn vim com- 
positam ex iis more consueto juxta [100] generalem constructionem numeri 205 ; sed 
etiam sic facilius idem praestatur ; centro Y intervallo cujusvis abscissae Ad figurae I in- 
veniatur in figura 24 sub littera A in recta CC' punctum d, sumaturque de versus Y 
aequalis ordinatae dh figuras i , ductoque ea perpendiculo in CA, erigatur eidem CA itidem 
perpendicularis dh dupla da versus plagam electam ad arbitrium pro attractionibus, vel 
versus oppositam, prout ilia ordinata in fig. I attractionem, vel repulsionem expresserit, 
& erit punctum h ad curvam exprimentem legem virium, qua punctum ubicunque 
collocatum in recta C'C solicitatur a binis X, Y. 



Demonstratio facilis est : si enim ducatur dX, & in ea sumatur dc aequalis de, 
ac compleatur rhombus debc ; patet fore ejus verticem b in recta dA secante angulum 
XdY bifariam, cujus diameter db exprimet vim compositam a binis de, dc, quae bifariam 
secaretur a diametro altera ec, & ad angulos rectos, adeoque in ipso illo puncto a ; & dh, 
dupla da, aequabitur db exprimenti vim, quae respectu A erit attractiva, vel repulsiva, prout 
ilia dh figurae I fuerit itidem attractiva, vel repulsiva. 

2I 7- Porro ex ipsa constructione patet, si centro Y, intervallis AE, AG, AI figuras i 
inveniantur in recta CAC' hujus figurae positae sub littera B puncta E, G, I, &c, ea fore 
limites respectu novae curvas ; & eodem pacto reperiri posse limites E', G', Y, &c. ex parte 
opposita A ; in iis enim punctis evanescente de figuras ejusdem positae sub A, evadit nulla 
da, & db. Notandum tamen, ibi in figura posita sub B mutari plagam attractivam in 

(p) Ducta enim LK in Fig. 23. ipsam FC secabit alicubi in I bifariam, W ad angulos rectos ex rhombi natura. 

Dicatur CD = x, CF = y, DB = a, W erit CB = Vaa + xx, fcf CD = *.CB = Vaa + xx : : CI = Jy.CK = 

\/aa + xx, quo valore posito in tequatione curvie figura I pro valore ordinata, y vaa + xx ffo valore abscissa, habebitur 
immediate cfquatio nova per x, y, W constants, qua ejusmodi curvam deUrminabit, 









& all the specific differences between the large number of bodies which they yield ; the 
agreement between the Theory & the whole of Nature is truly remarkable. But what 
has so far been said refers to masses, & to the application of the Theory to Physics. Before 
we come to this, however, I will discuss certain particular cases, out of an innumerable 
number of those which refer to the different laws concerning the action of two points on 
a third. 

214. If we wish to consider the laws that arise in the case of a straight line drawn The force exerted 
through D perpendicular to AB, or in the case of AB itself produced on either side, first by. two points on a 
of all it is easily seen that the direction of the resultant force in either case will coincide the?* straight* 1 line 
with the line itself without any lateral force or any declination from the straight line which Joining them, or in 
is drawn towards or away from D. In the case of AB itself the matter is self-evident ; whicifblsects it'at 
for the forces which pertain to the two points either have the same direction as one another, "8 ht angles. 

or are opposite in direction, since the third point lies in the same straight line as each of 
the two former points. Whence it comes about that the resultant force is equal to the 
sum, or the difference, of the two component forces ; & it will be in the same straight 
line as they. In the case of the line at right angles, the matter can be quite easily 
demonstrated. For, if in Fig. 23 the straight line DC were perpendicular to AB, passing 
through its middle point, then will AC, BC be equal to one another. Hence, the forces, 
by which C is influenced by A & B, will also be equal ; secondly, they will either be both 
attractive, as CL, CK, or they will be both repulsive, as CN, CM. Hence the resultant 
force, CF, or CH, will be the diagonal of a rhombus, & thus it will bisect the angle LCK, 
or NCM. Now since these angles are also bisected by the straight line DC, on account 
of the equality of the triangles DCA, DCB, it is evident that CF, CH must coincide with 
DC. Therefore, in these cases the perpendicular force FO, which was obtained in the 
two previous figures, will vanish. Also in these cases, the whole matter can be represented 
by a single equation (?) ; & the one, which refers to the latter case, can be found quite 

215. The law in the case of the straight line perpendicular to the straight line joining Construction for 
the two points, & equally distant from each, is graphically given in Fig. 24 ; to avoid * he curve . e iv ^s 

. . . ^ 1r .' . -r- i -i i r r .the law in the 

confusion the curve itself is given in rig. 243, whilst the construction for it is given separately second case. 

in Fig. 24A. These two figures are but one & the same, if the points X,Y,E,A,E' are 

supposed to be the same in both. Then, in the figure, X,Y are two points of matter, & 

the straight line CC' bisects XY at A. The curve, which here gives the resultant forces 

by means of the ordinates drawn to it, is constructed from that of Fig. i : & this can be 

done, by finding the forces for the points, each for each, then the force compounded from 

them in the usual manner according to the general construction given in Art. 205. But 

the same thing can be more easily obtained thus :, With centre Y, & radius equal to any 

abscissa Ad in Fig. i, construct a point d in the straight line CC', of Fig. 24A, & mark off 

de towards Y equal to the ordinate db in Fig. i ; draw ea perpendicular to CA, & erect 

a perpendicular, dh, to the same line CA also, so that dh = 2ae ; this perpendicular should 

be drawn towards the side of CA which is chosen at will to represent attractions, or towards 

the opposite side, according as the ordinate in Fig. i represents an attraction or a repulsion ; 

then the point h will be a point on the curve expressing the law of forces, with which a 

point situated anywhere on the line CC' will be influenced by the two points X & Y. 

. 216. The demonstration is easy. For, if dX is drawn, & in it dc is taken equal to de, Proof of the fore- 
& the rhombus debc is completed, then it is clear that the point b will fall on the straight gng construction, 
line dA. bisecting the angle X/Y ; & the diagonal of this rhombus represents the resultant 
of the two forces de, dc. Now, this diagonal is bisected at right angles by the other diagonal 
ec, & thus, at the point a in it. Also dh, being double of da, will be equal to db, which 
expresses the resultant force ; this will be attractive with respect to A, or repulsive, according 
as the ordinate dh in Fig. I is also attractive or repulsive. 

217. Further, from the construction, it is evident that, if with centre Y & radii Further properties 
respectively equal to AE, AG, AI in Fig. i, there are found in the straight line CAC' of sort. ' 
Fig. 248 the points E, G, I, &c, then these will be limit-points for the new curve ; & 
that in the same way limit-points E', G', I', &c. may be found on the opposite side of A. 
For, since at these points, in Fig. 24A, de vanishes, it follows that da & db become nothing 
also. Yet it must be noted that, in this case, in Fig. 248, there is a change from the attractive 

(p) For, if in Fig. 23, LK is drawn, it will cut FC somewhere, in I say ; & it will be at right angles to it 
on account of the nature of a rhombus. Sup-pose CD = x, CF = y, DB = a ; then CB = \/(a z + x 2 ), W toe have 

CD (or x) : CB (or ^(a* + x*) = CI (or Jy) : CK, /. CK = y.yV + x*)/2x ; 

y if this value is substituted in the equation of the curve in Fig. I instead of the ordinate, W ^/ (a z + x z ) for the 
abscissa, we shall get straightaway a new equation in x, y, \ constants ; & Ms will determine a curve of the kind 
under consideration. 


repulsivam, & vice versa ; nam in toto tractu CA vis attractiva ad A habet directionem 
CC', & in tractu AC' vis itidem attractiva ad A habet directionem oppositam C'C. Deinde 
facile patebit, vim in A fore nullam, ubi nimirum oppositae vires se destruent, adeoque 
ibi debere curvam axem secare ; ac licet distantiae AX, AY fuerint perquam exiguse, ut 
idcirco repulsiones singulorum punctorum evadant maximse ; tamen prope A vires erunt 
perquam exiguae ob inclinationes duarum virium ad XY ingentes, & contrarias ; & si ipsae 
AY, AX fuerint non majores, quam sit AE figurae I ; postremus arcus EDA erit repulsivus ; 
secus si fuerint majores, quam AE, & non majores, quam AG, atque ita porro ; cum vires 
in exigua distantia ab A debeant esse ejus directionis, quam in fig. I requirunt abscissas 
paullo majores, quam sit haec YA. Postrema crura T/>V,T"/>'V, patet, fore attractiva ; 
& si in figura I fuerint asymptotica, fore asymptotica etiam hie ; sed in A nullum erit 
asymptoticum crus. 

2l8> At curva C l uae ex hibet in fig. 25 legem virium pro recta CC' transeunte per duo 
casus prioris. puncta X, Y, est admodum diversa a priore. Ea facile construitur : satis est pro quovis 

ejus puncto d assumere in fig. I duas abscissae aequales, alteram Yd hujus figurae, alteram 
Xd ejusdem, & sumere hie db aequalem [101] summae, vel differentiae binarum ordinatarum 
pertinentium ad eas abscissas, prout fuerint ejusdem directionis, vel contrariae, & earn 
ducere ex parte attractiva, vel repulsiva, prout ambae ordinatae figurae I, vel earum major, 
attractiva fuerit, vel repulsiva. Habebitur autem asymptotus bYc, & ultra ipsam crus 
asymptoticum DE, citra ipsam autem crus itidem asymptoticum dg attractivum respectu 
A, cui attractivum, sed directionis mutatas respectu CC', ut in fig. superiore diximus, ad 
partes oppositas A debet esse aliud g'd', habens asymptotum c'V transeuntem per X ; 
ac utrumque crus debet continuari usque ad A, ubi curva secabit axem. Hoc postremum 
patet ex eo, quod vires oppositae in A debeant elidi ; illud autem prius ex eo, quod si a 
sit prope Y, & ad ipsum in infinitum accedat, repulsio ab Y crescat in infinitum, vi, quae 
provenit ab X, manente finita ; adeoque tam summa, quam differentia debet esse vis 
repulsiva respectu Y, & proinde attractiva respectu A, quae imminutis in infinitum distantiis 
ab Y augebitur in infinitum. Quare ordinata ag in accessu ad bYc crescet in infinitum ; 
unde consequitur, arcum gd fore asymptoticum respectu Yc ; & eadem erit ratio pro a'g', 
& arcu g'd' respectu b'Xc'. 

Ejus curvae pro- 219. Poterit autem etiam arcus curvae interceptus asymptotis bYc, b'Xc' sive cruribus 

at mutata 


mutata ^S> ^'g' secare alicubi axem, ut exhibet figura 26 ; quin immo & in locis pluribus, si nimirum 
distantia puncto- AY sit satis major, quam AE figurae i, ut ab Y habeatur alicubi citra A attractio, & ab X 

curva casus*aiterius! repulsio, vel ab X repulsio major, quam repulsio ab Y. Ceterum sola inspectione 
postremarum duarum figurarum patebit, quantum discrimen inducat in legem virium, vel 
curvam, sola distantia punctorum X, Y. Utraque enim figura derivata est a figura I, & in 
fig. 25 assumpta est XY sequalis AE figurae I, in fig. 26 aequalis AI, ejusdem quae variatio 
usque adeo mutavit figurse genitae ductum ; & assumptis aliis, atque aliis distantiis punc- 
torum X, Y, aliae, atque aliae curvae novae provenirent, quae inter se collatae, & cum illis, 
quae habentur in recta CAC' perpendiculari ad XAY, uti est in fig. 24 ; ac multo magis 
cum iis, quae pertinentes ad alias rectas mente concipi possunt, satis confirmant id, quod 
supra innui de tanta multitudine, & varietate legum provenientium a sola etiam duo- 
rum punctorum agentium in tertium dispositione diversa ; ut & illud itidem patet ex 
sola etiam harum trium curvarum delineatione, quanta sit ubique conformitas in arcu illo 
attractive TpV, ubique conjuncta cum tanto discrimine in arcu se circa axem contorquente. 

genera hujus 220. Verum ex tanto discriminum numero unum seligam maxime notatu dignum, 
Usima! ' g & maximo nobis usui futurum inferius. Sit in fig. 2jC 'AC axis idem, ac in fig. i, & quin- 

que arcus consequenter accept! alicubi GHI, IKL, LMN, NOP, PQR sint aequales 
prorsus inter se, ac similes. Ponantur autem bina puncta B', B hinc, & inde ab A in fig. 28 
[102] ad intervallum aequale dimidiae amplitudini unius e quinque iis arcubus, uti uni 
GI, vel IL ; in fig. 29 ad intervallum aequale integrae ipsi amplitudini ; in fig. 30 ad 
intervallum aequale duplae ; sint autem puncta L, N in omnibus hisce figuris eadem, & 
quaeratur, quae futura sit vis in quovis puncto g in intervallo LN in hisce tribus posi- 
tionibus punctorum B', B. 



1 7 o 



side to the repulsive side, & vice versa. For along the whole portion CA, the force of 
attraction towards A has the direction CC', whilst for the portion AC', the force of attraction 
also towards A has the direction C'C. Secondly, it will be clear,/ seen that the force at 
A will be nothing ; for there indeed the forces, being equal & opposite, cancel one another, 
& so the curve cuts the axis there ; & although the distances AX, AY would be very small, 
& thus the repulsions due to each of the two points would be Immensely great, nevertheless, 
close to A, the resultants would be very small, on account of the inclinations of the two 
forces to XY being extremely great & oppositely inclined. Also if AY, AX were not greater 
than AE in Fig. i, the last arc would be repulsive ; & attractive, if they were greater than 
AE, but not greater than AG, & so on ; for the forces at very small distances from A must 
have their directions the same as that required in Fig. I for abscissae that are slightly greater 
than YA. The final branches TpV, T'p'V will plainly be attractive ; &, if in Fig. i they 
were asymptotic, they would also be asymptotic in this case ; but there will not be an 
asymptotic branch at A. 

218. But the curve, in Fig. 25, which expresses the law of forces for the straight line Construction fo- 
CC', when it passes through the points X,Y, is quite different from the one just considered. j| the^aw^'tte 
It is easily constructed ; it is sufficient, for any point d upon it, to take, in Fig. i, two first case, 
abscissae, one equal to Yd, & the other equal to Xd ; & then, for Fig. 25, to take dh equal 

to the sum or the difference of the two ordinates corresponding to these abscissas, according 
as they are in the same direction or in opposite directions ; &, according as each ordinate, 
or the greater of the two, in Fig. i, is attractive or repulsive, to draw dh on the attractive 
or repulsive side of CC'. Moreover there will be obtained an atymptote bYc ; on the 
far side of this there will be an asymptotic branch DE, & on the near side of it there will 
also be an asymptotic branch dg, which will be attractive with respect to A ; & with respect 
to this part, there must be another branch g'd', which is attractive but, since the direction 
with regard to CC' is altered, as we mentioned in the case of the preceding figure, falling 
on the opposite side of CC' ; this has an asymptote c'b' passing through X. Also each 
branch must be continuous up to the point A, where it cuts thVaxis. This last fact is 
evident from the consideration that the equal & opposite forces at A must cancel one another ; 
& the former is clear from the fact that, if a is very near to Y, & approaches indefinitely 
near to it, the repulsion due to Y increases indefinitely, whilst the force due to X remains 
finite. Thus, both the sum & the difference must be repulsive with respect to Y, & therefore 
attractive with respect to A ; & this, as the distance from Y is diminished indefinitely, will 
increase indefinitely. Hence the ordinate ag, when approaching bYc, increases indefinitely : 
& it thus follows that the arc gd will be asymptotic with respect to Yc ; & the reasoning 
will be the same for a'g', & the arc g'd', with respect to b'Xc', 

219. Again, it is even possible that the arc intercepted between the asymptotes bYc, The properties of 
b'Xc', i.e., between the branches dg, d'g', to cut the axis somewhere, as is shown in Fig. 26 ; encescor^espo'n'dSg 
nay rather, it may cut it in more places than one, for instance, if AY is sufficiently greater to changed dis- 
than AE in Fig. i ; so that, at some place on the near side of A, there is obtained an attraction p^s ^"clrnpari 6 
from the point Y & a repulsion from the point X, or a repulsion from X greater than the son with the curve 
repulsion fiom Y. Besides, by a mere inspection of the last two figures, it will be evident other case. in the 
how great a difference in the law of forces, & the curve, may be derived from the mere 

distance apart of the points X & Y. For both figures are derived from Fig. I, &, in Fig. 25, 
XY is taken equal to AE in Fig. i , whilst, in Fig. 26, it is taken equal to AI of Fig. i ; & 
this variation alone has changed the derived figure to such a degree as is shown. If other 
distances, one after another, are taken for the points X & Y, fresh curves, one after the 
other, will be produced. If these are compared with one another, & with those that are 
obtained for a straight line CAC' perpendicular to XAY, like the one in Fig 24, nay, far 
more, if they are compared with those, referring to other straight lines, that can be imagined, 
will sufficiently confirm what has been said above with regard to the immense number & 
variety of the laws arising from a mere difference of disposition of the two points that act 
on the third. Also, from the drawing of merely these three curves, it is plainly seen 
what great uniformity there is in all cases for the attractive arc TpV, combined always 
with a great dissimilarity for the arc that is twisted about the axis. 

_ 220. But I will select, from this great number of different cases, one which is worth T ^ Tee classes of 
notice in a high degree, which also will be of the greatest service to us later. In Fig. 27, we u 
let CAC' be the same axis as in Fig. i, & let the five arcs, GHI, IKL, LMN, NOP, PQR 
taken consecutively anywhere along it, be exactly equal & like one another. Moreover, 
in Fig. 28, let the two points B & B', one on each side of A, be taken at a distance equal 
to half the width of one of these five arcs, i.e., half of the one GL, or LI ; in Fig. 29, at 
3. distance equal to the whole of this width ; &, in Fig. 30, at a distance equal to double 
the width ; also let the points L,N be the same in all these figures. It is required to find 
the force at any point g in the interval LN, for these three positions of the points B & B'. 

I 7 2 


Determinatip vis 
compositaa in iis- 

221. Si in Fig. 27 capiantur hinc, & inde ab ipso g intervalla sequalia intervallis AB', 
AB reliquarum trium figurarum ita, ut ge, gi respondeant figurae 28 ; gc, gm figures 29 ; 
ga, go figurae 30 ; patet, intervallum ei fore aequale amplitudini LN, adeoque Le, Ni 
aequales fore dempto communi Lz, sed puncta e, i debere cadere sub arcus proximos 
directionum contrariarum ; ob arcuum vero aequalitatem fore aequalem vim ef vi contrariae 
il, adeoque in fig. 28 vim ab utraque compositam, respondentem puncto g, fore nullam. 
At quoniam gc, gm integrae amplitudini aequantur ; cadent puncta c, m sub arcus IKL, 
NOP, conformes etiam directione inter se, sed directionis contraries respectu arcus LMN, 
eruntque asquales wzN, cl ipsi gL, adeoque attractiones mn, cd, & repulsioni gh aequales, 
& inter se ; ac idcirco in figura 29 habebitur vis attractiva gh composita ex iis binis dupla 
repulsivae figurae 27. Demum cum ga, go sint sequales duplae amplitudini, cadent puncta 
a, o sub arcus GHI, PQR conformis directionis inter se, & cum arcu LMN, eruntque 
pariter binae repulsiones ab, op aequales repulsioni gh, & inter se. Quare vis ex iis com- 
positae pro fig. 30 erit repulsio gh dupla repulsionis gh figurae 27, & aequalis attraction! 
figurae 29. 

vhn'in tractu*' 0116 222< ^^ igi tur j am patet, loci geometric! exprimentis vim compositam, qua bina 

tinuo nuiiam, in puncta B', B agunt in tertium, partem, quae respondet intervallo eidem LN, fore in prima 
aha attractionem, e tribus eorum positionibus propositis ipsum axem LN, in secunda arcum attractivum 

in aha repulsionem, T , ,,., . . , . i i , r 

manente distantia ; LMN, in tertia repulsivum, utroque reccdente ab axe ubique duplo plus, quam in fig. 
Ph y sica 27 ; ac pro quovis situ puncti g in toto intervallo LN in primo e tribus casibus fore prorsus 
nullam, in secundo fore attractionem, in tertio repulsionem aequalem ei, quam bina puncta 
B', B exercerent in tertium punctum situm in g, si collocarentur simul in A, licet in omnibus 
hisce casibus distantia puncti ejusdem g a medio systematis eorundem duorum punctorum, 
sive a centre particulae constantis iis duobus punctis sit omnino eadem. Possunt autem 
in omnibus hisce casibus puncta B', B esse simul in arctissimis limitibus cohaesionis inter 
se,- adeoque particulam quandam constantis positionis constituere. Aequalitas ejusmodi 
accurata inter arcus, & amplitudines, ac limitum distantias in figura I non dabitur uspiam ; 
cum nullus arcus curvae derivatae utique continuae, deductae nimirum certa lege a curva 
continua, possit congruere accurate cum recta ; at poterunt ea omnia ad sequalitatem 
accedere, quantum [103] libuerit ; poterunt haec ipsa discrimina haberi ad sensum per 
tractus continues aliis modis multo adhuc pluribus, immo etiam pluribus in immensum, 
ubi non duo tantummodo puncta, sed immensus eorum numerus constituat massulas, 
quae in se agant, & ut in hoc simplicissimo exemplo deprompto e solo trium punctorum 
systemate, multo magis in systematis magis compositas, & plures idcirco variationes admit- 
tentibus, in eadem centrorum distantia, pro sola varia positione punctorum componentium 
massulas ipsas vel a se mutuo repelli, vel se mutuo attrahere, vel nihil ad sensum agere in 
se invicem. Quod si ita res habet, nihil jam mirum accidet, quod quaedam substantial 
inter se commixtse ingentem acquirant intestinarum partium motum per effervescentiam, 
& fermentationem, quas deinde cesset, particulis post novam commixtionem respective 
quiescentibus ; quod ex eodem cibo alia per secretionem repellantur, alia in succum 
nutrititium convertantur, ex quo ad eandem prseterfluente distantiam alia aliis partibus 
solidis adhaereant, & per alias valvulas transmittantur, aliis libere progredientibus. Sed 
adhuc multa supersunt notatu dignissima, quae pertinent ad ipsum etiam adeo simplex 
trium punctorum systema. 

Alius casus vis nul- 
lius trium puncto- 
rum positorum in 
directum e x dis- 
tantiis limitum : 
tres alii in quorum 
binis vis nulla ex 
elisione contrari- 

223. Jaceant in figura 31 tria puncta A,D,B, in directum : ea poterunt respective 
quiescere, si omnibus mutuis viribus careant, quod fieret, si tres distantiae AD, DB, AB 
omnes essent distantiae limitum ; sed potest haberi etiam quies respectiva per elisionem 
contrariarum virium. Porro virium mutuarum casus diversi tres esse poterunt : vel enim 
punctum medium D ab utroque extremorum A, B attrahitur, vel ab utroque repellitur, 
vel ab altero attrahitur, ab altero repellitur. In hoc postremo casu, patet, non haberi 
quietem respectivam ; cum debeat punctum medium moveri versus extremum attrahens 
recedendo simul ab altero extremo repellente. In reliquis binis casibus poterit utique 





B'A B 

FIG. 28. 


B' A B 

FIG. 29. 









L N 

FIG. 30. 




R C 


FIG. 27. 

L n. ," 5 

B'A B 

FIG. 28. 

B' A B 

B 7 

FIG .29. 

B L N 

FIG. 30. 


221. If, in Fig. 27, we take, on either side of this point g, intervals that are equal to Determination of 
the intervals AB', AB of the other three figures ; so that ge, gi correspond to Fig. 28 ; 

gc, gm to Fig. 29 ; & ga, go to Fig. 30 ; then it is plain that the interval ei will be equal 
to the width LN, & thus, taking away the common part Lz, we have L & Ni equal to one 
another, but the points e & i must fall under successive arcs of opposite directions. Now, 
on account of the equality of the arcs, the force ef will be equal to the opposite force il ; 
thus, in Fig. 28, the force compounded from the two, corresponding to the point g, will 
be nothing. Again, in Fig. 29, since gc, gm are each equal to the whole width of an arc, 
the points c & m fall under arcs IKL, NOP, which lie in the same direction as one another, 
but in the opposite direction to the arc LMN. Hence, mN, c\ will be equal to gL ; & 
thus the attractions mn, cd will be equal to the repulsion gb, & to one another. Therefore, 
in Fig. 29, we shall have an attractive force, compounded of these two, which is double 
of the repulsive force in Fig. 27. Lastly, in Fig. 30, since ga, go are equal to double the 
width of an arc, the points a & o will fall beneath arcs GHI, PQR, lying in the same direction 
as one another, & as that of the arc LMN as well. As before, the two repulsions, ab, op 
will be equal to the repulsion gb, & to one another. Hence, in Fig. 30, the force compounded 
from the two of them will be a repulsion gh which is double of the repulsion gh in Fig. 27, 
& equal to the attraction in Fig. 29. 

222. Therefore, from the preceding article, it is now evident that the part of the in one arrange- 

geometrical locus representing the resultant force, with which two points B', B act " '* 

upon a third, corresponding to the same interval LN, will be the axis LN itself in the first no force at ail, in 

of the three stated positions of the points ; in the second position it will be an attractive attraction" 5 * ^a 

arc LMN, & in the third a repulsive arc ; each of these will recede from the axis at all third a repulsion 

points along it to twice the corresponding distance shown in Fig. 27. So, for any position maining^constant 

of the point g in the whole interval LN, the force will be nothing at all in the first of the this result is of the 

three cases, an attraction in the second, & a repulsion in the third. This latter will be -physics. l 

equal to that which the two points B', B would exert on the third point, if they were both 

situated at the same time at the point A. And yet, in all these three cases, the distance 

of the point g under consideration remains absolutely the same, measured from the centre 

of the system of the same two points, or from the mean centre of a particle formed from 

them. Moreover, in all three cases, the points B',B may be in the positions defining the 

strongest limits of cohesion with regard to one another, & so constitute a particle fixed 

in position. Now we never can have such accurate equality as this between the arcs, the 

widths, & the distances of the limit-points ; for no arc of the derived curve, which is every- 

where continuous because it is obtained by a given law from a continuous curve, can possibly 

coincide accurately with a straight line ; but there could be an approximation to equality 

for all of them, to any degree desired. The same distinctions could be obtained, 

approximately for continuous regions in very many more different ways, nay the number 

of ways is immeasurable ; in which the number of points constituting the little masses 

is not two only, but a very large number, acting upon one another ; &, as in this very simple 

case derived from a consideration of a single system of three points, so, much more in systems 

that are more complex & on that account admitting of more variations, corresponding to 

a single variation of the points composing the masses, whilst the distance between the 

masses themselves remains the same, there may be either mutual repulsion, mutual attraction, 

or no mutual action to any appreciable extent. But, that being the case, there is 

nothing wonderful in the fact that certain substances, when mixed together, acquire a 

huge motion of their inmost parts, as in effervescence & fermentation ; this motion ceasing 

& the particles attaining relative rest after rearrangement. There is nothing wonderful 

in the fact that from the same food some things are repelled by secretion, whilst others 

are converted into nutritious juices ; & that from these juices, though flowing past at 

exactly the same distances, some things adhere to some solid parts & some to others ; that 

some are transmitted through certain little passages, some through others, whilst some 

pass along uninterruptedly. However, there yet remain many things with regard to this ever 

so simple system of three points ; & these are well worth our attention. 

223. In Fig. 31, let A,D,B be three points in a straight line. These will be at rest Another instance 

i 11 i n i ., , of no force in the 

with regard to one another if they lack all mutual forces ; & this would be the case, it the case of three points 
three distances AD, DB, AB were all distances corresponding to limit-points. In addition, sit V?;, t f. d i " a 

i 111 i i ,...*- in f ft. straight line at the 

relative rest could be obtained owing to elimination of equal & opposite iorces. .further, distances corre- 
there will be three different cases with regard to the mutual forces. For, either the middle spending to Hmit- 

. -i r i i > T> 11 ii i r V. points. Ihree 

point D is attracted by each of the outside points A & B, or it is repelled by each of them, others, in two of 
or it is attracted by one of them & repelled by the other. In the last case, it is evident ^^res^tenl'farce 
that relative rest could not obtain ; for the middle point must then be moved towards the arises from an eii- 
outside point that is attracting it, & recede from the other outside point which is repelling 
it at the same time. But in the other two cases, it is at least possible that there may be 


In eorum altero 
nisus ad recuper- 
andam positionem, 
in altero ad magis 
ab ea recedendum, 
si incipiant inde 

res haberi : nam vires attractive, vel repulsive, quas habet medium punctum, possunt 
esse aequales ; turn autem extrema puncta debebunt itidem attrahi a medio in primo casu, 
repelli in secundo ; quae si se invicem e contrario aeque repellant in casu primo, attrahant 
in secundo ; poterunt mutuse vires elidi omnes. 

224. Adhuc autem ingens est discrimen inter hosce binos casus. Si nimirum puncta 
ilia a directione rectae lineae quidquam removeantur, ut nimirum medium punctum D 
distet jam non nihil a recta AB, delatam in C, in secundo casu adhuc magis sponte recedet 
inde, & in primo accedet iterum ; vel si vi aliqua externa urgeatur, conabitur recuperare 
positionem priorem, & ipsi urgenti vi resi- 
stet. Nam binae repulsiones CM, CN adhuc 
habebuntur in secundo casu in ipso primo 
recessu a D (licet ese mutatis jam satis distan- 
tiis BD, AD inBC, AC, evadere possint at- 
tractiones) & vim com-[i opponent direc- 
tam per CH contrariam directioni tendenti 
ad rectam AB. At in primo casu habebuntur 
attractiones CL, CK, quae component vim 
CF directam versus AB, quo casu attractio 
AP cum repulsione AR, et attractio BV, 
cum repulsione BS component vires AQ, BT, 
quibus puncta A, B ibunt obviam puncto C 
redeunti ad rectam transituram per illud 

T"V /"* 

FIG. 31. 

Theoria generalior 
indicata : t r i u m 
punctorum jacen- 
tium in directum : 
vis maxima ad 
conservandam dis- 


R A 

B S 

punctum E, quod est in triente rectae DC, 
& de quo supra mentionem fecimus num. 205. 

225. Haec Theoria generaliter etiam non rectilineae tantum, sed & cuivis position! 
trium massarum applicari potest, ac applicabitur infra, ubi etiam generale simplicissimum, 
ac fcecundissimum theorema eruetur pro comparatione virium inter se : sed hie interea 
evolvemus nonnulla, quae pertinent ad simpliciorem hunc casum trium punctorum. In- 
primis fieri utique potest, ut ejusmodi tria puncta positionem ad sensum rectilineam 
retineant cum prioribus distantiis, utcunque magna fuerit vis, quae ilia dimovere tentet, 
vel utcunque magna velocitas impressa fuerit ad ea e suo respectivo statu deturbanda. 
Nam vires ejusmodi esse possunt, ut tarn in eadem directione ipsius rectas, quam in 
directione ad earn perpendicular!, adeoque in quavis obliqua etiam, quae in eas duas resolvi 
cogitatione potest, validissimus exurgat conatus ad redeundum ad priorem locum, ubi inde 
discesserint puncta. Contra vim impressam in directione ejusdem rectae satis est, si pro 
puncto medio attractio plurimum crescat, aucta distantia ab utrolibet extreme, & plurimum 
decrescat eadem imminuta ; ac pro utrovis puncto extreme satis est, si repulsio decrescat 
plurimum aucta distantia ab extreme, & attractio plurimum crescat, aucta distantia a 
medio, quod secundum utique fiet, cum, ut dictum est, debeat attractio medii in ipsum 
crescere, aucta distantia. Si haec ita se habuerint, ac vice versa ; differentia virium vi 
extrinsecae resistet, sive ea tenet contrahere, sive distrahere puncta, & si aliquod ex iis 
velocitatem in ea directione acquisiverit utcunque magnam, poterit differentia virium 
esse tanta, ut extinguat ejusmodi respectivam velocitatem tempusculo, quantum libuerit, 
parvo, & post percursum spatiolum, quantum libuerit, exiguum. 

Quid ubi vis exter- 

& virgae flexiiis. 

226. Quod si vis urgeat perpendiculariter, ut punctum medium D moveatur 
per rectam DC perpendicularem ad AB ; turn vires CK, CL possunt utique esse ita validae, 
ut vis composita CF sit post recessum, quantum libuerit, exiguum satis magna ad ejusmodi 
vim elidendam, vel ad extinguendam velocitatem impressam. In casu vis, quas constanter 
urgeat, & punctum D versus C, & puncta A, B ad partes oppositas, habebitur inflexio ; ac 
in casu vis, quae agat in eadem directione rectae jungentis puncta, habebitur contractio, 
seu distractio ; sed vires resistentes ipsis poterunt esse ita validae, ut & inflexio, & contractio, 
vel distractio, sint prorsus insensibiles ; [105] ac si actione externa velocitas imprimatur 
punctis ejusmodi, quae flexionem, vel contractionem, aut distractionem inducat, turn 
ipsa puncta permittantur sibi libera ; habebitur oscillatio quasdam, angulo jam in alteram 
plagam obverso, jam in alteram oppositam, ac longitudine ejus veluti virgae constantis iis 
tribus punctis jam aucta, jam imminuta, fieri poterit ; ut oscillatio ipsa sensum omnem 
effugiat, quod quidem exhibebit nobis ideam virgae, quam vocamus rigidam, & solidam, 
contractionis nimirum, & dilatationis incapacem, quas proprietates nulla virga in Natura 

[The reader should draw a more general figure for Art. 224 & 227, taking AD, DB 
unequal and CD not at right angles to AB.] 


relative rest ; for the attractive, or repulsive, forces which are acting on the middle point 
may be equal. But then, in these cases, the outside points must be respectively attracted, 
or repelled by the middle point ; & if they are equally & oppositely repelled by one another 
in the first case, & attracted by one another in the second case, then it will be possible for 
all the mutual forces to cancel one another. 

224. Further, there is also a very great difference between these two cases. For in one of these 
instance, if the points are moved a small distance out of the direct straight line, so that endeavour 6 towards 
the middle point D, say, is now slightly off the straight line AB, being transferred to C, a recovery of posi- 
then, if left to itself, it will recede still further from it in the first case, & will approach t^warts^iurther 
it once more in the second case. Or, if it is acted on by some external force, it will endeavour recession from it, 
to recover its position & will resist the force acting on it. For two repulsions, CM, CN, moved o 

will at first be obtained in the second case, at the first instant of motion from the position position. 
D ; although indeed these may become attractions when the distances BD, AD are 
sufficiently altered into the distances BC, AC. These will give a resultant force 
acting along CH in a direction away from the straight line AB. But in the first 
case we shall have two attractions CL, CK ; & these will give a force directed 
towards AB. In this case, the attraction AP combined with the repulsion AR, & 
the attraction BV combined with the repulsion BY, will give resultant forces, AQ, BT, 
under the action of which the points A,B will move in the opposite direction to that of 
the point C, as it returns to the straight line passing through that point E, which is a third 
of the way along the straight line DC, of which mention was made above in Art. 205. 

225. This Theory can also be applied more generally, to include not only a position Enunciation of a 
of the three points in a straight line but also any position whatever. This application more general theory 
will be made in what follows, where also a general theorem, of a most simple & fertile nature ly^g ^ a straight 
will be deduced for comparison of forces with one another. But for the present we will line ; possibility of 
consider certain points that have to do with this more simple case of three points. First tendSg^conser^ 
of all, it may come about that three points of this kind may maintain a position practically vation of distance, 
in a straight line, no matter how great the force tending to drive them from it may be, 

or no matter how great a velocity may be impressed upon them for the purpose of disturbing 
them from their relative positions. For there may be forces of such a kind that both in 
the direction of the straight line, & perpendicular to it, & hence in any oblique direction 
which may be mentally resolved into the former, there may be produced an extremely 
strong endeavour towards a return to the initial position as soon as the points had departed 
from it. To counterbalance the force impressed in the direction of the same straight line 
itself, it is sufficient if the attraction for the middle point should increase by a large amount 
when the distance from either of the outside points is increased, & should be decreased 
by a large amount if this distance is decreased. For either of the outside points it is sufficient 
if the repulsion should greatly decrease, as the distance is increased, from the outside point, 
and the attraction should greatly increase, as the distance is increased, from the middle 
point ; & this second requirement will be met in every case, since, as has been said, and 
attraction on it of the middle point will necessarily increase when the distance is increased. 
If matters should turn out to be as stated, or vice versa, then the difference of the forces 
will resist the external force, whether it tries to bring the points together or to drive them 
apart ; & if any one of them should have acquired a velocity in the direction of the straight 
line, no matter how great, there will be a possibility that the difference of the forces may 
be so great that it will destroy any relative velocity of this kind, in any interval of time, 
no matter how short the time assigned may be ; & this, after passing over any very small 
assigned space, no matter how small. 

226. But if the force acts perpendicularly, so that, for instance, the point D moves what happens if 
along the line DC perpendicular to AB, then the forces CK, CL, can in any case be so ^es n^tTct ah^g 
strong that the resultant force CF may become, after a recession of any desired degree the straight line ; 
of smallness, large enough to eliminate any force of this kind, or to destroy any impressed 

velocity. In the case of a force continually urging the point D towards C, & the points 
A & B in the opposite direction, there will be some bending ; & in the case of a force acting 
in the same direction as the straight line joining the two points, there will be some contraction 
or distraction. But the forces resisting them may be so strong that the bending, the 
contraction, or the distraction will be altogether inappreciable. If by external action a 
velocity is impressed on points of this kind, & if this induces bending, contraction or distraction, 
& if the points are then left to themselves, there will be produced an oscillation, in which 
the angle will jut out first on one side & then on the other side ; & the length of, so to 
speak, the rod consisting of the three points will be at one time increased & at another 
decreased ; & it may possibly be the case that the oscillation will be totally unappreciable ; 
& this indeed will give us the idea of a rod, such as we call rigid & solid, incapable of 
being contracted or dilated ; these properties are possessed by no rod in Nature perfectly 


I 7 8 


habet accurata tales, sed tantummodo ad sensum. Quod si vires sint aliquanto debiliores, 
turn vero & inflexio ex vi externa mediocri, & oscillatio, ac tremor erunt majores, & jam 
hinc ex simplicissimo trium punctorum systemate habebitur species quaedam satis idonea 
ad sistendum animo discrimen, quod in Natura observatur quotidie oculis. inter virgas 
rigidas, ac eas, quae sunt flexiles, & ex elasticitate trementes. 

Systemate inflexo 227. Ibidem si binse vires, ut AQ, BT fuerint perpendiculares ad AB, vel etiam 

vls r ^ncti^medii utcunque parallels inter se, tertia quoque erit parallela illis, & aequalis earum summae, 
contraria extremis, sed directionis contrariae. Ducta enim CD parallela iis, turn ad illam KI parallela BA, 
" ' erit ob CK > y B sequales, triangulum CIK aequale simili BTV, sive TBS, adeoque CI squalls 

BT, IK aequalis BS, sive AR, vel QP. Quare si sumpta IF aequali AQ ducatur KF ; erit 
triangulum FIK aequale AQP, ac proinde FK aequalis, & parallela AP, sive LC, & CLFK 
parallelogrammum, ac CF, diameter ipsius, exprimet vim puncti C utique parallelam 
viribus AQ, BT, & asqualem earum summae, sed directionis contrariae. Quoniam vero 
est SB ad BT, ut BD ad DC ; ac AQ ad AR, ut DC ad DA ; erit ex aequalitate perturbata 
AQ ad BT, ut BD ad DA, nimirum vires in A, & B in ratione reciproca distantiarum AD, 
DB a recta CD ducta per C secundum directionem virium. 



Postremum theo- 
rema generate, ubi 
etiam tria puncta 
non jaceant in di- 

Equilibrium trium 
punctorum non in 
directum jacentium 
impossible sine vi 
externa, nisi sint 
in distantiis limi- 
tum : cum iis qui 
nisus ad retinen- 
dam formam syste- 

228. Ea, quas hoc postremo numero demonstravimus, aeque pertinent ad actiones 
mutuas trium punctorum habentium positionem mutuam quamcunque, etiam si a rectilinea 
recedat quantumlibet ; nam demonstratio generalis est : sed ad massas utcunque inaequales, 
& in se agentes viribus etiam divergentibus, multo generalius traduci possunt, ac traducentur 
inferius, & ad aequilibrii leges, & vectem, & centra oscillationis ac percussionis nos deducent. 
Sed interea pergemus alia nonnulla persequi pertinentia itidem ad puncta tria, quae in 
directum non jaceant. 

229. Si tria puncta non jaceant in directum, turn vero sine externis viribus non poterunt 
esse in aequilibrio ; nisi omnes tres distantiae, quae latera trianguli constituunt, sint dis- 
tantiae limitum figurae i. Cum enim vires illae mutuae non habeant [106] directiones 
oppositas ; sive unica vis ab altero e reliquis binis punctis agat in tertium punctum, sive 
ambae ; haberi debebit in illo tertio puncto motus, vel in recta, quae jungit ipsum cum 
puncto agente, vel in diagonali parallelogrammi, cujus latera binas illas exprimant vires. 
Quamobrem si assumantur in figura I tres distantiae limitum ejusmodi, ut nulla ex iis sit 
major reliquis binis simul sumptis, & ex ipsis constituatur triangulum, ac in singulis angu- 
lorum cuspidibus singula materiae puncta collocentur ; habebitur systema trium punctorum 
quiescens, cujus punctis singulis si imprimantur velocitates aequales, & parallelae ; habebitur 
systema progrediens quidem, sed respective quiescens ; adeoque istud etiam systema 
habebit ibi suum quemdam limitem, sed horum quoque limitum duo genera erunt : ii, 
qui orientur ab omnibus tribus limitibus cohaesionis, erunt ejusmodi, ut mutata positione, 
conentur ipsam recuperare, cum debeant conari recuperare distantias : ii vero, in quibus 
etiam una e tribus distantiis fuerit distantia limitis non cohaesionis, erunt ejusmodi, ut 
mutata positione : ab ipsa etiam sponte magis discedat systema punctorum eorundem. Sed 
consideremus jam casus quosdam peculiares, & elegantes, & utiles, qui hue pertinent. 

Eiegans theoria 230. Sint in fig. 32 tria puncta A,E,B ita collocata, ut tres distantise AB, AE, BE sint 
ratto eiupsTs binis distantiae limitum cohaesionis, & postremae duae 
aiiis occupantibus sint aequales. Focis A, B concipiatur ellipsis 

transiens per E, cujus axis transversus sit FO, 

conjugatus EH, centrum D : sit in fig. I AN 

aequalis semiaxi transverse hujus DO, sive BE, 

vel AE, ac sit DB hie minor, quam in fig. I 

amplitude proximorum arcuum LN, NP, & sint 

in eadem fig. i arcus ipsi NM, NO similes, & 

aequales ita, ut ordinatae uy, zt, aeque distantes 

ab N, sint inter se aequales. Inprimis si punctum 

materiae sit hie in E ; nullum ibi habebit vim, 

cum AE, BE sint aequales distantiae AN limitis 

N figurae I ; ac eadem est ratio pro puncto 

collocate in H. Quod si fuerit in O, itidem 

erit in aequilibrio. Si enim assumantur in fig. I 

Az, AM aequales hisce BO, AO ; erunt Nz, 

foco : vis nulla in 
verticibus axium. 

illius aequales DB, DA hujus, adeoque & inter se. Quare & vires illius zt, uy erunt aequales 
inter se, quae cum pariter oppositae directionis sint, se mutuo elident ; ac eadem ratio est 
pro collocatione in F. Attrahetur hie utique A, & repelletur B ab O ; sed si limes, qui 
respondet distantiae AB, sit satis validus ; ipsa puncta nihil ad sensum discedent a focis 


accurately, but only approximately. But if the forces are somewhat more feeble, then 
indeed, under the action of a moderate external force, the bending, the oscillation & the 
vibration will all be greater ; & from this extremely simple system of three points we now 
obtain several kinds of cases that are adapted to giving us a mental conception of the differences, 
that meet our eyes every day in Nature, between rigid rods & those that are flexible & 
elastically tremulous. 

227. At the same time, if the two forces, represented by AQ, BT, were perpendicular In * s y stem dis - 

A T> n i i_ ^i. ^L- j f u f i n i torted by parallel 

to AB, or parallel to one another in any manner, then the third force would also be parallel forces the force on 
to them, equal to their sum, but in the opposition direction. For, if CD is drawn parallel the middle point is 
to the forces, & KI parallel to BA to meet CD in I, then, since CK & VB are equal to direction t thaTof 
one another, the triangle CIK will be equal to the similar triangle BTV, or to the triangle the outside forces, 
TBS ; & therefore CI will be equal to BT, IK to BS or AR or QP. Hence if IF is taken sumT* 
equal to AQ & KF is drawn, then the triangle FIK will be equal to AQP, & thus FK 
will be equal parallel to AP or LC, CLFK will be a- parallelogram, & its diagonal CF 
will represent the force for the point C, in every case parallel to the forces AQ, BT, & 
equal to their sum, but opposite in direction. But, because SB : BT : : BD : DC, & 
AQ : AR : : DC : DA ; hence, ex cequali we have AQ : BT : : BD : DA, that is to say, the 
forces on A & B are in the inverse ratio of the distances AD & DB, drawn from the straight 
line CD in the direction of the forces. 

228. What has been proved in the last article applies equally to the mutual actions The last theorem in 
of three points having any relative positions whatever, even if it departs from a rectilinear fhe tiTree^point^do 
position to any extent you may please. For the demonstration is general ; &, further, the not He in a straight 
results can be deduced much more generally for masses that are in every manner unequal, line- 

& that act upon one another even with diverging forces ; & they will be thus deduced 
later ; & these will lead us to the laws of equilibrium, the lever, & the centres of oscillation 
& percussion. But meanwhile we will go straight on with our consideration of some 
matters relating in the same manner to three points, which do not lie in a straight line. 

229. If the three points do not lie in a straight line, then indeed without the presence Equilibrium of 
of an external force they cannot be in equilibrium ; unless all three distances, which form ^o^no* ^ in^a 
the sides of the triangle, are those corresponding to the limit-points in Fig. I. For, since straight line isim- 
the mutual forces do not have opposite directions, either a single force from one of the F^^^^^ 1 ^ ^ 

. i i i i TT i iri presence 01 an 

remaining two points acts on the third, or two such forces. Hence there must be for that external force, 
third point some motion, either in the direction of the straight line joining it to the acting "? B less 3l . th H is l^ 

' o j o o are at distances 

point, or along the diagonal of the parallelogram whose sides represent those two forces, corresponding to 

Therefore, if in Fig. i we take three limit-distances of such a kind, that no one of them is 8 ' 

in his 
greater than the other two taken together, & if from them a triangle is formed & at each case, to 'conserve 

vertical angle a material point is situated, then we shall have a system of three points at rest, sygte^" 11 of the 

If to each point of the system there is given a velocity, and these are all equal & parallel to one 

another, we shall have a system which moves indeed, but which is relatively at rest. Thus 

also that system will have a certain limit of its own ; moreover, of such limits there are also 

two kinds. Namely, those that arise from all three limit-points being those of cohesion 

which will be such that, if the relative position is altered, they will strive to recover it ; 

for they are bound to try to restore the distances. Secondly, those in which one of the 

three distances corresponds to a limit-point of non-cohesion, which will be such that, if 

the relative position is altered, the system will of its own accord depart still more from it. 

However, let us now consider certain special cases, that are both elegant & useful, for which 

this is the appropriate place. 

230. In Fig. 32, let the three points A,E,B be so placed that the three distances AB, An elegant theory 
AE, BE correspond to limit-points of cohesion, & let the two last be equal to one another. i^the'periineter of 
Suppose that an ellipse, whose foci are A & B, passes through E ; let the transverse axis of an ellipse, each of 
this be FO, & the conjugate axis EH, & the centre D. In Fig. i, let AN be equal to behVpiaced irTa 
the transverse semiaxis DO of Fig. 32, that is equal to BE or AE ; also in the latter figure focus ; no force at 
let DB be less than the width of the successive arcs LN, NP of Fig. i ; also, in Fig. i, let en 
the arcs NM, NO be similar & equal, so that the ordinates uy, zt, which are equidistant 
from N, are equal to one another. Then, first of all, if in Fig. 32, the point of matter 
is situated at E, there will be no force upon it ; for AE, BE are equal to the distance AN 
of the limit-point N in Fig. i ; & the argument is the same for a point situated at H. 
Further, if it is at O, it will in like manner be in equilibrium. For, if in Fig. i we take 
Az, Au equal to AO, BO of Fig. 32, then Nz, NM of the former figure will be equal to DB, 
DA of the latter ; & thus equal also to one another. Hence also the forces in that figure, 
zt & uy, will be equal to one another ; & since they are likewise opposite in direction, they 
will cancel one another ; & the argument is the same for a point situated at F. Here 
in every case A is attracted & B is repelled from O ; but if the limit-point, which corresponds 
to the distance AB is strong enough, the points will not depart to any appreciable extent 



In reliquis puncti 
perimetri vis direc- 
ta per ipsam peri- 
metrum versus ver- 
tices axis conju- 

Analogia verticum 
binorum axium 
cum limitibus cur- 
vae virium. 

Quando limites 
contrario m o d o 
positi : casus ele- 
gantissimi alterna- 
tionis p 1 u r i u m 
limitum in peri- 
metro ellipseos. 


ellipseos, in quibus fuerant collocata, vel si debeant discedere ob limitem minus validum, 
considerari poterunt per externam vim ibidem immota, ut contemplari liceat solam 
relationem tertii puncti ad ilia duo. 

231. Manet igitur immotum, ac sine vi, 
punctum collocatum tarn in verticibus axis con- 
jugati ejus ellipseos, quam in verticibus axis 
transversi ; & si ponatur in quovis puncto C 
[107] perimetri ejus ellipseos, turn ob AC, CB 
simul aequales in ellipsi axi transverse, sive duplo 
semiaxi DO ; erit AC tanto longior, quam ipsa 
DO, quanto BC brevior ; adeoque si jam in fig. 
I sint AM, Az aequales hisce AC, BC ; habe- 
buntur ibi utique uy, zt itidem aequales inter se. 
Quare hie attractio CL sequabitur repulsioni 
CM, & LIMC erit rhombus, in quo inclinatio 1C 
secabit bifariam angulum LCM ; ac proinde si 
ea utrinque producatur in P, & Q ; angulus ACP, 
qui est idem, ac LCI, erit aequalis angulo BCQ, 
qui est ad verticem oppositus angulo ICM. Quse 
cum in ellipsi sit notissima proprietas tangentis 

relatae ad focos ; erit ipsa PQ tangens. Quamobrem dirigetur vis puncti C in latus secundum 
tangentem, sive secundum directionem arcus elliptici, atque id, ubicunque fuerit punctum 
in perimetro ipsa, versus verticem propiorem axis conjugati, & sibi relictum ibit per ipsam 
perimetrum versus eum verticem, nisi quatenus ob vim centrifugam motum non nihil 
adhuc magis incurvabit. 

232. Quamobrem hie jam licebit contemplari in hac curva perimetro vicissitudinem 
limitum prorsus analogorum limitibus cohaesionis, & non cohaesionis, qui habentur in axe 
rectilineo curvae primigeniae figures I. Erunt limites quidam in E, in F, in H, in O, in 
quibus nimirum vis erit nulla, cum in omnibus punctis C intermediis sit aliqua. Sed in 

E, & H erunt ejusmodi, ut si utravis ex parte punctum dimoveatur, per ipsam perimetrum, 
debeat redire versus ipsos ejusmodi limites, sicut ibi accidit in limitibus cohaesionis ; at in 

F, & O erit ejusmodi, ut in utramvis partem, quantum libuerit, parvum inde punctum 
dimotum fuerit, sponte debeat inde magis usque recedere, prorsus ut ibi accidit in limitibus 
non cohaesionis. 

233. Contrarium accideret, si DO aequaretur distantiae limitis non cohaesionis : turn 
enim distantia BC minor haberet attractionem CK, distantia major AC repulsionem CN, 
& vis composita per diagonalem CG rhombi CNGK haberet itidem directionem tangentis 
ellipseos ; & in verticibus quidem axis utriusque haberetur limes quidam, sed punctum 
in perimetro collocatum tenderet versus vertices axis transversi, non versus vertices axis 
conjugati, & hi referrent limites cohaesionis, illi e contrario limites non cohaesionis. Sed 
adhuc major analogia in perimetro harum ellipsium habebitur cum axe curvae primigeniae 
figurae I ; si fuerit DO asqualis distantiae limitis cohaesionis AN illius, & DB in hac major, 
quam in fig. i amplitude NL, NP ; multo vero magis, si ipsa hujus DB superet plures 
ejusmodi amplitudines, ac arcuum aequalitas maneat hinc, & inde per totum ejusmodi 
spatium. Ubi enim AC hujus figurae fiet aequalis abscissae AP illius, etiam BC hujus fiet 
pariter aequalis AL illius. Quare in ejusmodi loco habebitur limes, & ante ejusmodi locum 
versus A distantia [108] longior AC habebit repulsionem, & BC brevior attractionem, 
ac rhombus erit KGNC, & vis dirigetur versus O. Quod si alicubi ante in loco adhuc 
propriore O distantiae AC, BC aequarentur abscissis AR, AI figurae i ; ibi iterum esset 
limes ; sed ante eum locum rediret iterum repulsio pro minore distantia, attractio pro 
majore, & iterum rhombi diameter jaceret versus verticem axis conjugati E. Generaliter 
autem ubi semiaxis transversus aequatur distantiae cujuspiam limitis cohaesionis, & distantia 
punctorum a centre ellipseos, sive ejus eccentricitas est major, quam intervallum dicti 
limitis a pluribus sibi proximis hinc, & inde, ac maneat aequalitas arcuum, habebuntur in 
singulis quadrantibus perimetri ellipeos tot limites, quot limites transibit eccentricitas 
hinc translata in axem figurae I, a limite illo nominato, qui terminet in fig. i semiaxem 
transversum hujus ellipseos ; ac praetererea habebuntur limites in verticibus amborum 
ellipseos axium ; eritque incipiendo ab utrovis vertice axis conjugati in gyrum per ipsam 
perimetrum is limes primus cohaesionis, turn illi proximus esset non cohaesionis, deinde 


from the foci of the ellipse, in which they were originally situated ; or, if they are forced 
to depart therefrom owing to the insufficient strength of the limit-point, they may be 
considered to be kept immovable in the same place by means of an external force, so that 
we may consider the relation of the third point to those two alone. 

2i>i. A point, then, which is situated at one of the vertices of the conjugate axis of At remaining points 

, ,. J . , ' , r i_ i o j -L of tne perimeter 

the ellipse or at one of the vertices of the transverse axis remains motionless & under the the force directed 
action of no force. If it is placed at any point C in the perimeter of the ellipse, then, since alon s the perimeter 

A ^i /~.T> i i IT i i Till .is towards the ver- 

AC, CB taken together are m the ellipse equal to the transverse axis, or double the semi- tices of the conju- 

axis DO, AC will be as much longer than DO as BC is shorter. Hence, if in Fig. i AM, 8 ate axis - 

Az are equal to these lines AC, BC, we shall have in every case, in Fig. I, uy, zt also equal 

to one another. Therefore, in Fig. 32, the attraction CL will be equal to the repulsion 

CM, & LIMC will be a rhombus, in which the inclination 1C will bisect the angle LCM. 

Hence if it is produced on either side to P & Q, the angle ACP, which is the same as the 

angle LCI will be equal to the angle BCQ, which is vertically opposite to the angle ICM. 

Now this is a well-known property with respect to the tangent referred to the foci in the 

case of an ellipse ; & therefore PQ is the tangent. Hence the force on the point C is directed 

laterally along the tangent, i.e., in the direction of the arc of the ellipse ; & this is true, 

no matter where the point is situated on the perimeter, & the force is towards the nearest 

vertex of the conjugate axis ; if left to itself, the point will travel along the perimeter 

towards that vertex, except in so far as its motion is disturbed somewhat in addition, owing 

to centrifugal force. 

232. Hence we can consider in this curved perimeter the alternation of limit-points Analogy between 
as being perfectly analogous to those of cohesion & non-cohesion, which were obtained in two ^xes 63 * the 
the rectilinear axis of the primary curve of Fig. I. There will be certain limit-points at limit-points of the 
E, F, H, O, in which there is no force, whilst in all intermediate points such as C there c 

will be some force. But at E & H they will be such that, if the point is moved towards 
either side along the perimeter, it must return towards such limit-points, just as it has to 
do in the case of limit-points of cohesion in Fig. I. But at F & O, the limit-point would 
be such that, if the point is moved therefrom to either side by any amount, no matter 
how small, it must of its own accord depart still further from it ; exactly as it fell out in 
Fig. i for the limit-points of non-cohesion. 

233. Just the contrary would happen, if DO were equal to the distance corresponding when the limit 
to a limit-point of non-cohesion. For then the smaller distance BC would have an P omi f are disposed 

ATT- i T A ^i i /^XT i i r -i , in the opposite 

attraction CK, & the greater distance AC a repulsion CJN ; the resultant force along the way ; most elegant 
diagonal CG of the rhombus CNGK would in the same way have its direction along the instances of aiter- 

,,,. . . r -i i 111 .,..9 nation of several 

tangent to the ellipse, & at the vertices of either axis there would be certain limit-points ; limit-points in the 

but a point situated in the perimeter would tend towards the vertices of the transverse g^ 6 *" of the 

axis, & not towards the vertices of the conjugate axis ; & the latter are of the nature of 

limit-points of cohesion & the former of non-cohesion. However, a still greater analogy 

in the case of the perimeter of these ellipses with the axis of the primary curve of Fig. i 

would be obtained, if DO were taken equal to the distance corresponding to the limit-point 

of cohesion AN in that figure, & in the present figure DB were taken greater than the 

width of NL, NP in Fig. i ; much more so, if DB were greater than several of these widths, 

& the equality between the areas on one side & the other held good throughout the whole 

of the space taken. For where AC in the present figure becomes equal to the abscissa AP 

of the former, BC in the present figure will likewise become equal to AL in the former. 

Hence at a position of this kind there will be a limit-point ; & before a position of this 

kind, towards O, the longer distance AC will have a repulsion & the shorter distance BC 

an attraction, KGNC will be a rhombus, & the force will be directed towards O. But if 

at some position, on the side of O, & still nearer to O, the distances AC, BC were equal 

to the abscissae AR, AI of Fig. I, then again there would be a limit-point ; but before 

that position there would return once more a repulsion for the smaller distance & an 

attraction for the greater, & once more the diagonal of the rhombus would lie in the direction 

of E, the vertex of the conjugate axis. Moreover, in general, whenever the transverse 

semiaxis is equal to the distance corresponding to any limit-point of cohesion, & the distance 

of the points from the centre of the ellipse, i.e., its eccentricity, is greater than the interval 

between the said limit-point & several successive limit-points on either side of it, & the 

equality of the arcs holds good, then for each quadrant of the perimeter of the ellipse there 

will be as many limit-points as the number of limit-points in the axis of Fig. I that the 

eccentricity will cover when transferred to it from the present figure, measured from that 

limit-point mentioned as terminating in Fig. I the transverse semiaxis of the ellipse of the 

present figure ; in addition there will be limit-points at the vertices of both axes of the 

ellipse. Beginning at either vertex -of the conjugate axis, & going round the perimeter, 

the first limit-point will be one of cohesion, then the next to it one of non-cohesion, then 


alter cohaesionis, & ita porro, donee redeatur ad primum, ex quo incceptus fuerit gyrus, 
vi in transitu per quemvis ex ejusmodi limitibus mutante directionem in oppositam. Quod 
si semiaxis hujus ellipseos aequetur distantiae limitis non cohsesionis figurae i ; res ecdem 
ordine pergit cum hoc solo discrimine, quod primus limes, qui habetur in vertice semiaxis 
conjugati sit limes non cohaesionis, turn eundo in gyrum ipsi proximus sit cohsesionis limes, 
deinde iterum non cohaesionis, & ita porro. 

Perimetn piunum 2 , . Verum est adhuc alia quaedam analogia cum iis limitibus ; si considerentur 

elhpsium aequiva- JT .. . .... , . .. '..... 

lentes limitibus. plures ellipses nsdem illis iocis, quarum semiaxes ordine suo aequentur distantns, in altera 
cujuspiam e limitibus cohaesionis figuras I, in altera limitis non cohaesionis ipsi proximi, 
& ita porro alternatim, communis autem ilia eccentricitas sit adhuc etiam minor quavis 
amplitudine arcuum interceptorum limitibus illis figurse I, ut nimirum singulae ellipsium 
perimetri habeant quaternos tantummodo limites in quatuor verticibus axium. Ipsae 
ejusmodi perimetri totae erunt quidam veluti limites relate ad accessum, & recessum a 
centro. Punctum collocatum in quavis perimetro habebit determinationem ad motum 
secundum directionem perimetri ejusdem ; at collocatum inter binas perimetros diriget 
semper viam suam ita, ut tendat versus perimetrum definitam per limitem cohaesionis 
figurae I, & recedat a perimetro definita per limitem non cohaesionis ; ac proinde punctum 
a perimetro primi generis dimotum conabitur ad illam redire ; & dimotum a perimetro 
secundi generis, sponte illam adhuc magis fugiet, ac recedet. 

Demonstrate. 235. Sint enim in fig. 33. ellipsium FEOH, F'E'O'H', F"E"O"H" semiaxes DO, 

D'O', D"O" aequales primus di-[iO9]-stantiae AL limitis non cohaesionis figurae i ; secundus 
distantiae AN limitis cohaesionis ; tertius distantiae AP limitis iterum non cohaesionis, & 
primo quidem collocetur C aliquanto ultra perimetrum mediam F'E'O'H' : erunt AC, 
BC majores, quam si essent in perimetro, adeoque in fig. I factis AM, Az majoribus, quam 
essent prius, decrescet repulsio zt, crescet attractio uy ; ac proinde hie in parallelogram mo 
LCMI erit attractio CL major, quam repulsio CM, & idcirco accedet directio diagonalis 
CI magis ad CL, quam ad CM, & inflectetur introrsum versus perimetrum mediam. 
Contra vero si C' sit intra perimetrum mediam, factis BC', AC' minoribus, quam si essent 
in perimetro media ; crescet repulsio C'M', & decrescet attractio C'L', adeoque directio 
C'l' accedet magis ad priorem C'M', quam ad posteriorem C'L', & vis dirigetur extrorsum 
versus eandem mediam perimetrum. Contrarium autem accideret ob rationem omnino 
similem in vicinia primae vel tertias perimetri : atque inde patet, quod fuerat propositum. 

blematum s e g e s, 
sed minus utilis : 
immensa combina- 
tionum varietas. 

Alias curvas eiiip- 236. Quoniam arcus hinc, & inde a quovis limite non sunt prorsus aequales ; quanquam, 

das 13 ; ampfa^pro- ut su P ra observavimus num. 184, exigui arcus ordinatas ad sensum aequales hinc, & inde 
habere debeant ; curva, per cujus tangentem perpetuo dirigatur vis, licet in exigua eccen- 
tricitate debeat esse ad sensum ellipsis, tamen nee in iis erit ellipsis accurate, nee in 
eccentricitatibus majoribus ad ellipses multum accedet. Erunt tamen semper aliquae 
curvae, quae determinent continuam directionem virium, & curvse etiam, quae trajectoriam 
describendam definiant, habita quoque ratione vis centifugae : atque hie quidem uberrima 
seges succrescit problematum Geometrise, & Analysi exercendae aptissimorum ; sed omnem 
ego quidem ejusmodi perquisitionem omittam, cujus nimirum ad Theoriae applicationem 
usus mihi idoneus occurrit nullus ; & quae hue usque vidimus, abunde sunt ad ostendendam 
elegantem sane analogiam alternationis in directione virium agentium in latus, cum virium 
primigeniis simplicibus, ac harum limitum cum illarum limitibus, & ad ingerendam animo 
semper magis casuum, & combinationum diversarum ubertatem tantam in solo etiam 
trium punctorum systemate simplicissimo ; unde conjectare liceat, quid futurum sit, ubi 
immensus quidam punctorum numerus coalescat in massulas constituentes omnem hanc 
usque adeo inter se diversorum corporum multitudinem sane immensam. 

Conversio t o t i u s 
systematis illaesi : 
impulsu per peri- 
metrum ellip sees 
oscil latio: idea 
liquationis, & con- 

237. At praeterea est & alius insignis, ac magis determinatus fructus, quern ex ejusmodi 
contemplationibus capere possumus, usui futurus etiam in applicatione Theoriae ad 
Physicam. Si nimirum duo puncta A, & B sint in distantia limitis cohaesionis satis validi, 
& punctum tertium collocatum in vertice axis conjugati in E distantiam a reliquis habeat, 
quam habet limes itidem cohaesionis satis validus ; poterit sane [no] vis, qua ipsum 
retinetur in eo vertice, esse admodum ingens pro utcunque exigua dimotione ab eo loco, 



FIG. 33. 

i8 4 


FIG. 33. 


another of cohesion, & so on, until we arrive at the first of them, from which the circuit 
was commenced ; & the force changes direction as we pass through each of the limit-points 
of this kind to the exactly opposite direction. But if the semiaxis of this ellipse is equal 
to the distance corresponding to a limit-point of non-cohesion in Fig. i, the whole matter 
goes on as before, with this difference only, namely, that the first limit-point at the vertex 
of the conjugate semiaxis becomes one of non-cohesion ; then, as we go round, the next 
to it is one of cohesion, then again one of non-cohesion, & so on. 

234. Now there is yet another analogy with these limit-points. Let us consider a The perimeters of 
number of ellipses having the same foci, of which the semiaxes are in order equal to the sev< r ral ellipses 

,. ,....,-,. , - . . equivalent to limit- 

distances corresponding to limit-points in .tig. I, namely to one of cohesion for one, to points. 

that of non-cohesion next to it for the second, & so on alternately ; also suppose that the 
eccentricity is still smaller than any width of the arcs between the limit-points of Fig. I, 
so that each of the elliptic perimeters has only four limit-points, one at each of the four 
vertices of the axes. The whole set of such perimeters will be somewhat of the nature 
of limit-points as regards approach to, or recession from the centre. A point situated in 
any one of the perimeters will have a propensity for motion along that perimeter. If it 
is situated between two perimeters, it will always direct its force in such a way that it will 
tend towards a perimeter corresponding to a limit-point of cohesion in Fig. i, & will 
recede from a perimeter corresponding to a limit-point of non-cohesion. Hence, if a point 
is disturbed out of a position on a perimeter of the first kind, it will endeavour to return 
to it ; but if disturbed from a position on a perimeter of the second kind, it will of its 
own accord try to get away from it still further, & will recede from it. 

235. In Fig. 33, of the semiaxes DO, DO', DO" of the ellipses FEOH, F'E'O'H', Demonstration. 
F"E"O"H". let the first be equal to the distance corresponding to AL, a limit-point of 
non-cohesion in Fig. I, the second to AN, one of cohesion, the third to AP, one of non- 
cohesion. In the first place, let the point C be situated somewhere outside the middle 
perimeter F'E'O'H' ; then AC, BC will be greater than if they were drawn to the perimeter. 

Hence, in Fig. I, since AM, Az would be made greater than they were formerly, the repulsion 
zt would decrease, & the attraction uy would increase. Therefore, in Fig. 33, in the 
parallelogram LCMI, the attraction CL will be greater than the repulsion CM, & so the 
direction of the diagonal CI will approach more nearly to CL than to CM, & will be turned 
inwards towards the middle perimeter. On the other hand, if C' is within the middle 
perimeter, BC', AC' are made smaller than if they were drawn to the middle perimeter ; 
the repulsion C'M' will increase, & the attraction C'L' will decrease, & thus the direction 
of CT will approach more nearly to the former, C'M', than to the latter, C'L' ; & the 
force will be directed outwards towards the middle perimeter. Exactly the opposite would 
happen in the neighbourhood of the first or third perimeter, & the reasoning would be 
similar. Hence, the theorem enunciated is evidently true. 

236. Now, since the arcs on either side of any chosen limit-point are not exactly equal, other curves to 
& yet, as has been mentioned above in Art. 184, very small arcs on either side are bound b ,f. substltuted for 

i IT i 11 i ellipses ; an ample 

to nave approximately equal ordmates ; the curve, along the tangent to which the force crop of theorems, 

is continually directed, although for small eccentricity it must be practically an ellipse, b "g a " ot v^|et h US oi 

yet will neither be an ellipse accurately in this case, nor approach very much to the form combinations. 

of an ellipse for larger eccentricity. Nevertheless, there will always be certain curves 

determining the continuous direction of the force, & also curves determining the path 

described when account is taken of the centrifugal force. Here indeed there will spring 

up a most bountiful crop of problems well-adapted for the employment of geometry & 

analysis. But I am going to omit all discussion of that kind ; for I can find no fit use for 

them in the application of my Theory. Also those which we have already seen are quite 

suitable enough to exhibit the truly elegant analogy between the alternation in direction 

of forces acting in a lateral direction & the simple primary forces, between the limit-points 

of the former & those of the latter ; also for impressing on the mind more & more the 

great wealth of cases & different combinations to be met with even in the single very simple 

system of three points. From this it may be conjectured what will happen when an 

immeasurable number of points coalesce into small masses, from which are formed all that 

truly immense multitude of bodies so far differing from one another. 

237. In addition to the above, there is another noteworthy & more determinate result Rotation of the 
to be derived from considerations of this kind, & one that will be of service in the application ** ' e sy ., s , t m 

t 1 -r>i T.I i-i ! -i A n T> T T mtact ) oscillation 

oi the 1 neory to rnysics. r or instance, it the two points A & is are at a distance corresponding along the perimeter 
to a limit-point of cohesion that is sufficiently strong, & the third point situated at the of the ellipse due to 

T-, , . . . , . ' 1-1 an impulse ; the 

vertex r, oi the conjugate axis is at a distance from the other two which corresponds to idea of liquefaction 
a limit-point of cohesion that is also sufficiently strong, then the force retaining the point & congelation, 
at that vertex might be great enough, for any slight disturbance from that position, to 
prevent it from being moved any further, unless through the action of a huge external 


ut sine ingenti externa vi inde magis dimoveri non possit. Turn quidem si quis impediat 
motum puncti B, & circa ipsum circumducat punctum A, ut in fig. 34 abeat in A' ; abibit 
utique & E versus E', ut servetur forma trianguli AEB, quam necessario requirit conver- 
satio distantiarum, sive laterum inducta a limitum validitate, & 
in qua sola poterit respective quiescere systema, ac habebitur 
idea quaedam soliditatis cujus & supra injecta est mentio. At 
si stantibus in fig. 32 punctis A, B per quaspiam vires externas, 
quae eorum motum impediant, vis aliqua exerceatur in E ad 
ipsum a sua positione deturbandum ; donee ea fuerit medio- 
cris, dimovebit illud non nihil ; turn, ilia cessante, ipsum se resti- 
tuet, & oscillabit hinc, & inde ab illo vertice per perimetrum 
curvae cujusdam proximse arcui elliptico. Quo major fuerit vis 
externa dimovens, eo major oscillatio net ; sed si non fuerit 
tanta, ut punctum a vertice axis conjugati recedens deveniat ad 
verticem axis transversi ; semper retro cursus reflectetur, & de- 
scribetur minus, quam semiellipsis. Verum si vis externa coegerit 
percurrere totum quadrantem, & transilire ultra verticem axis 
transversi ; turn verogyrabit punctum circumquaque per totam FIG. 34. 

perimetrum motu continue, quern a vertice axis conjugati ad 

verticem transversi retardabit, turn ab hoc ad verticem conjugati accelerabit, & ita porro, 
nee sistetur periodicus conversionis motus, nisi exteriorum punctorum impedimentis 
occurrentibus, quae sensim celeritatem imminuant, & post ipsos ejusmodi motus periodicos 
per totum ambitum reducant meras oscillationes, quas contrahant, & pristinam debitam 
positionem restituant, in qua una haberi potest quies respectiva. An non ejusmodi aliquid 
accidit, ubi solida corpora, quorum partes certam positionem servant ad se invicem, ingenti 
agitatione accepta ab igneis particulis liquescunt, turn iterum refrigescentes, agitatione 
sensim cessante per vires, quibus igneae particulae emittuntur, & evolant, positionem prio- 
rem recuperant, ac tenacissime iterum servant, & tuentur ? Sed haec de trium punctorum 
systemate hucusque dicta sint satis. 
Systema punctorum 238. Quatuor, & multo magis plurium, punctorum systemata multo plures nobis 

quatuor, in eodem . . J , .. . . ,',..., . r -p, 

piano cum distan- vanationes objicerent ; si rite ad examen vocarentur ; sed de us id unum innuam. H,a 
tiis hmitum, suao q u id e m in piano eodem possunt positionem mutuam tueri tenacissime ; si singulorum 

forma; tenax. f. . r ,. . ....... . ,. , ,. 

distantiae a reliquis sequentur distantns hmitum satis validorum tigurae I : neque emm 
in eodem piano positionem respectivam mutare possunt, aut aliquod ex iis exire e piano 
ducto per reliqua tria, nisi mutet distantiam ab aliquo e reliquis, cum datis trium punctorum 
distantiis mutuis detur triangulum, quod constituere debent, turn datis distantiis quarti 
a duobus detur itidem ejus positio respectu eorum in eodem piano, & detur distantia ab 
eorum tertio, quae, si id punctum exeat e [in] priore piano, sed retineat ab iis duobus 
distantiam priorem, mutari utique debet, ut facili negotio demonstrari potest. 

Alia ratio system- 239. Quin immo in ipsa ellipsi considerari possunt puncta quatuor, duo in focis, & 

quatuor ^^eodem a ^ a ^ uo nmc > & ' m< ^ e a vertice axis conjugati in ea distantia a se invicem, ut vi mutua 

piano cum idea repulsiva sibi invicem elidant vim, qua juxta praecedentem Theoriam urgentur in ipsum 

flexiiis "Systema verticem ; quo quidem pacto rectangulum quoddam terminabunt, ut exhibet fig. 35, in 

eorundem forms punctis A, B, C, D. Atque inde si supra angulos quadratae basis assurgant series ejusmodi 

nes^rifmrticufa- punctorum exhibentium series continuas rectangulorum, habebitur quaedam adhuc magis 

rum pyramidaiium. praecisa idea virgae solidae, in qua si basis ima inclinetur ; statim omnia superiora puncta 

movebuntur in latus, ut rectangulorum illorum positionem retineant,& celeritas conversionis 

erit major, vel minor, prout major fuerit, vel minor vis ilia in latus, quae ubi fuerit aliquanto 

languidior, multo serius progredietur vertex, quam fundum, . 

& inflectetur virga, quae inflexio in omni virgarum genere 

apparet adhuc multo magis manifesta, si celeritas conversionis C O 

fuerit ingens. Sed extra idem planum possunt quatuor puncta 
collocata ita, ut positionem suam validissime tueantur, etiam 
ope unicae distantiae limitis unici satis validi. Potest enim fieri 
pyramis regularis, cujus latera singula triangularia habeant 

ejusmodi distantiam. Turn ea pyramis constituet particulam 

quandam suae figurae tenacissimam, quae in puncta, vel pyra- /\ 3 

mides ejusmodi aliquanto remotiores ita poterit agere, ut ejus FlG 35 

puncta respectivum situm nihil ad sensum mutent. Ex quatuor 

ejusmodi particulis in aliam majorem pyramidem dispositis fieri poterit particula secundi 
ordinis aliquanto minus tenax ob majorem distantiam particularum primi earn componen- 


force. In that case, if the motion of the point B were prevented, & the point A were set 
in motion round B, so that in Fig. 34 it moved to A', then the point E would move off 
to E' as well, so as to conserve the form of the triangle AEB, as is required by the conservation 
of the sides or distances which is induced by the strength of the limits ; & the system can 
be relatively at rest in this form only ; thus we get an idea of a certain solidity, of which 
casual mention has" already been made above. But if, in Fig. 2, whilst the points A,B, 
are kept stationary by means of an external force preventing their motion, some force is 
exerted on the point at E to disturb it from its position, then, as long as the force is only 
moderate, it will move the point a little ; & afterwards, when the force ceases, the point 
will recover its position, & will then oscillate on each side of the vertex along a perimeter 
of the curve that closely approximates to an elliptic arc. The greater the external force 
producing the motion, the greater the oscillation will be ; but if it is not so great as to make 
the point recede from the vertex of the conjugate axis until it reaches the vertex of the 
transverse axis, its path will always be retraced, & the arc described will be less than a 
semi-ellipse. But if the external force should compel the point to traverse a whole quadrant 
& pass through the vertex of the transverse axis, then indeed the point will make a complete 
circuit of the whole perimeter with a continuous motion ; this will be retarded from the 
vertex of the conjugate axis to that of the transverse axis, then accelerated from there 
onwards to the vertex of the conjugate axis, & so on ; there will not be any periodic reversal 
of motion, unless there are impediments met with from external points that appreciably 
diminish the speed ; in which case, following on such periodic motions round the whole circuit, 
there will be a return to mere oscillations ; & these will be shortened, & the original position 
restored, the only one in which there can possibly be relative rest. Probably something 
of this sort takes place, when solid bodies whose parts maintain a definite position with 
regard to one another, if subjected to the enormous agitation produced by fiery particles, 
liquefy ; & once more freezing, as the agitation practically ceases on account of forces due 
to the action of which the fiery particles are driven out & fly off, recover their initial position 
& again keep & preserve it most tenaciously. But let us be content with what has been said 
above with regard to a system of three points for the present. 

238. Systems of four, & much more so for more, points would yield us many more varia- A system of four 
tions, if they were examined carefully one after the other ; but I will only mention one thing jj^^ces^orre* 
about such systems. It is possible that such systems, in one plane, may conserve their rela- spending to Hmit- 
tive positions very tenaciously, if the distances of each from the rest are equal to the dis- ^ves^tslform. 0011 
tances in Fig. i corresponding to limit-points of sufficient strength. For neither can they 

change their relative position in the plane, nor can any one of them leave the plane drawn 
through the other three ; since, if the distances of three points from one another is given, 
then we are given the triangle which they must form ; & then being given the distances 
of the fourth point from two of these, we are also given the position of this fourth point 
from them, & therefore also the distance from the third of them. If the point should 
depart from the plane mentioned, & yet preserve its former distances from the two points 
the distance from the third point must be changed in any case, as can be easily proved. 

239. Again, we may consider the case of four points in the ellipse, two being at the A further consider- 
foci, & the other two on either side of a vertex of the conjugate axis at such a distance from a { io " ou f ^i^f 6 , 
one another, that the mutual repulsive force between them will cancel the force with which connection with 
they are urged towards that vertex, according to the preceding theorem. Thus, they are the idea of rigid & 

f i . T->- i A T, V-i T-V flexible rods; a. 

at the vertices of a rectangle, as is shown in rig. 35, where they occupy the points A,B,U,D. system of four 
Hence, if we have a series of points of this kind to stand above the four angles of the quadratic P mts m the fo . rm 

r i 11 i r i ^ of a pyramid; 

base, so as to represent continuous series of rectangles, we shall obtain from this supposition different arrange- 
a more precise idea than hitherto has been possible of a solid rod, in which, if the lowest ments f particular 

r r v i n T i 1-1 pyramids. 

set or points is inclined, all the points above are immediately moved sideways, but 
so that they retain the positions in their rectangles ; & the speed of rotation will be greater 
or less according as the force sideways was greater or less ; even where this force is somewhat 
feeble, the top will move considerably later than the base & the rod will be bent ; & the 
amount of bending in every kind of rod will be still more apparent if the speed of rotation is 
very great. Again, four points not in the same plane can be so situated that they preserve 
their relative position very tenaciously ; & that too, when we make use of but a single 
distance corresponding to a limit-point of sufficient strength. For they can form a regular 
pyramid, of which each of the sides of the triangles is of a length equal to this distance. 
Then this pyramid will constitute a particle that is most tenacious as regards its form ; 
& this will be able to act upon points, or pyramids of the same kind, that are more remote, 
in such a manner that its points do not alter their relative position in the slightest degree 
for all practical purposes. From four particles of this kind, arranged to form a larger 
pyramid, we can obtain a particle of the second order, somewhat less tenacious of form on 
account of the greater distance between the particles of the first order that compose it ; 


tium, qua fit, ut vires in easdem ab externis punctis impressae multo magis inaequales inter se 
sint,^quam fuerint in punctis constituentibus particulas ordinis primi ; ac eodem pacto ex 
his secundi ordinis particulis fieri possunt particulse ordinis tertii adhuc minus tenaces figurae 
suae, atque ita porro, donee ad eas deventum sit multo majores, sed adhuc multo magis 
mobiles, atque variabiles, ex quibus pendent chemica; operationes, & ex quibus haec ipsa 
crassiora corpora componuntur, ubi id ipsum accideret, quod Newtonus in postrema Optics 
questione proposuit de particulis suis primigeneis, & elementaribus, alias diversorum ordi- 
num particulas efformantibus. Sed de particularibus hisce systematis determinati punc- 
torum numeri jam satis, ac ad massas potius generaliter considerandas faciemus gradum. 

Transitus ad 240. In massis primum nobis se offerunt considerandas elegantissimse sane, ac foccund- 

massas : quid cen- m **t* ' j T->I 

trum gravitatis : issimae, & utilissimae propnetates centn gravitatis, quse quidem e nostra 1 heona sponte 
theoremata hk de propemodum fluunt, aut saltern eius ope evidentissime demonstrantur. Porro centrum 

eo demonstrando. L *. . .., . . . . . , . 

gravitatis a gravium aequihbno nomen accepit suum, a quo etiam ejus consideratio ortum 

duxit ; sed id quidem a gravi-[ii2]-tate non pendet, sed ad massam potius pertinet. 
Quamobrem ejus definitionem proferam ab ipsa gravitate nihil omnino pendentem, quan- 
quam & nomen retinebo, & innuam, unde originem duxerit ; turn demonstrabo accuratissime, 
in quavis massa haberi aliquod gravitatis centrum, idque unicum, quod quidem passim 
omittere solent, & perperam ; deinde ad ejus proprietatem praecipuam exponendam 
gradum faciam, demonstrando celeberrimum theorema a Newtono propositum, centrum 
gravitatis commune massarum, sive mihi punctorum quotcunque, & utcunque disposi- 
torum, quorum singula moveantur sola inertiae vi motibus quibuscunque, qui in singulis 
punctis uniformes sint, in diversis utcunque diversi, vel quiescere, vel moveri uniformiter 
in directum : turn vero mutuas actiones quascunque inter puncta quaelibet, vel omnia 
simul, nihil omnino turbare centri communis gravitatis statum quiescendi vel movendi 
uniformiter in directum, unde nobis & actionis, ac reactionis aequalitas in massis quibusque, 
& principia collisiones corporum definientia, & alia plurima sponte provenient. Sed 
aggrediamur ad rem ipsam. 

Definitio centri 241. Centrum igitur commune gravitatis punctorum quotcunque. & utcunque 

gravitatis non j- n f -j j j i 

pendens ab idea dispositorum, appellabo id punctum, per quod si ducatur planum quodcunque ; summa 

gravitatis : ejus distantiarum perpendicularium ab eo piano punctorum omnium jacentium ex altera 
idea 8 communi. C * ejusdem parte, sequatur summa distantiarum ex altera. Id quidem extenditur ad quas- 
cunque, & quotcunque massas ; nam eorum singulae punctis utique constant, & omnes 
simul sunt quaedam punctorum diversorum congeries. Nomen traxit ab aequilibrio 
gravium, & natura vectis, de quibus agemus infra : ex iis habetur illud, singula 
pondera ita connexa per virgas inflexiles, ut moveri non possint, nisi motu circa aliquem 
horizontalem axem, exerere ad conversionem vim proportionalem sibi, & distantiae perpen- 
diculari a piano verticali ducto per axem ipsum ; unde fit, ut ubi ejusmodi vires, vel, ut 
ea vocant, momenta virium hinc, & inde asqualia fuerint, habeatur aequilibrium. Porro 
ipsa pondera in nostris gravibus, in quibus gravitatem concipimus, ac etiam ad sensum 
experimur, proportionalem in singulis quantitati materiae, & agentem directionibus inter 
se parallelis, proportionalia sunt massis, adeoque punctorum eas constituentium numero ; 
quam ob rem idem est, ea pondera in distantias dncere, ac assumere summam omnium 
distantiarum omnium punctorum ab eodem piano. Quod si igitur respectu aggregati 
cujuscunque punctorum materiae quotcunque, & quomodocunque dispositorum sit aliquod 
punctum spatii ejusmodi, ut, ducto per ipsum quovis piano, summa distantiarum ab illo 
punctorum jacentium ex parte altera aequetur summse distantiarum jacentium ex altera ; 
concipiantur autem singula ea puncta animata viribus aequalibus, & parallelis, cujusmodi sunt 
vires, quas in nostris gravibus concipimus ; illud utique consequitur, [113] suspense utcunque 
ex ejusmodi puncto, quale definivimus gravitatis centrum, omni eo systemate, cujus 
systematis puncta viribus quibuscunque, vel conceptis virgis inflexibilus, & gravitate 
carentibus, positionem mutuam, & respectivum statum, ac distantias omnino servent, id 
systema fore in aequilibrio ; atque illud ipsum requiri, ut in aequilibrio sit. Si enim vel 
unicum planum ductum per id punctum sit ejusmodi, ut summae illae distantiarum non 
sint aequales hinc, & inde ; converse systemate omni ita, ut illud planum evadat verticale, 
jam non essent aequales inter se summae momentorum hinc, & inde, & altera pars alteri 
prseponderaret. Verum haec quidem, uti supra monui, fuit occasio quaedam nominis 
imponendi ; at ipsum punctum ea lege determinatum longe ulterius extenditur, quam 


for from this fact it comes about that the forces impressed upon these from external points 
are much more unequal to one another. than they would be for the points constituting 
particles of the first order. In the same manner, from these particles of the second order 
we might obtain particles of the third order, still less tenacious of form, & so on ; until 
at last we reach those which are much greater, still more mobile, & variable particles, which 
are concerned in chemical operations ; & to those from which are formed the denser bodies, 
with regard to which we get the very thing set forth by Newton, in his last question in 
Optics, with respect to his primary elemental particles, that form other particles of different 
orders. We have now, however, said enough concerning these systems of a definite number 
of points, & we will proceed to consider masses rather more generally. 

240. In dealing with masses, the first matters that present themselves for our considera- Passing on to 
tion are certain really very elegant, as well as most fertile & useful properties of the centre of "ntr? of "gravity^ 
gravity. These indeed come forth almost spontaneously from my Theory, or at least are Theorems to be 
demonstrated most clearly by means of it. Further, the centre of gravity derived its name 

from the equilibrium of heavy (gravis) bodies, & the first results in connection with the former 
were developed by means of the latter ; but in reality it does not depend on gravity, but rather is 
related to masses. On this account, I give a definition of it, which in no way depends on 
gravity, although I will retain the name, & will mention whence it derived its origin. Then 
I will prove with the utmost rigour that in every body there is a centre of gravity, & one 
only (a thing which is usually omitted by everybody, quite unjustifiably). Then I will 
proceed to expound its chief property, by proving the well-known theorem enunciated by 
Newton ; that the centre of gravity of masses, or, in my view, of any number of points in 
any positions, each of which is moved in any manner by the force of inertia alone, this 
force being uniform for the separate points but maybe non-uniform to any extent for 
different points, will be either at rest or will move uniformly in a straight line. Finally, 
I will show that any mutual action whatever between the points, or all of them taken 
together, will in no way disturb the state of rest or of uniform motion in a straight line of 
the centre of gravity. From which the equality of action & reaction in all bodies, & the 
principles governing the collision of solids, & very many other things will arise of them- 
selves. However let us set to work on the matter itself. 

241. Accordingly, I will call the common centre of gravity of any number of points, Definition of the 
situated in any positions whatever, that point which is such that, if through it any plane in^tndent^fTn 7 
is drawn, the sum of the perpendicular distances from the plane of all the points lying on idea of gravitation ; 
one side of it is equal to the sum of the distances of all the points on the other side of it. ^ dtfaiSorT 
The definition applies also to masses, of any sort or number whatever ; for each of the the usual idea, 
latter is made up of points, & all of them taken together are certain groups of different 

points. The name is taken from the equilibrium of weights (gravis), & from the principle 
of the lever, with which we shall deal later. Hence we obtain the principle that each of 
the weights, connected together by rigid rods in such a manner that the only motion possible 
to them is one round a horizontal axis, will exert a turning force proportional to itself & 
to its perpendicular distance from a vertical plane drawn through this axis. From which 
it comes about that, when the forces of this sort (or, as they are called, the moments of the 
forces) are equal to one another on this side & on that, then there is equilibrium. Further, 
the weights in our heavy bodies, in which we conceive the existence of gravity (& indeed 
find by experience that there is such a thing) proportional in each to the quantity of matter, 
& acting in directions parallel to one another, are proportional to the masses, & thus to 
the number of points that go to form them. Therefore, the product of the weights into 
the distances comes to the same